(covers information from several alternate timelines)
Maxwell Forrest was a Human Starfleet flag officer in charge of the NX Project during the 2140s and 2150s. As a commodore, he supervised the first tests of the warp five engine. As such, he worked closely with the Vulcans, who were supervising Humanity's progress into space.
As a Starfleet commodore, Forrest was responsible for overseeing the NX Project and did so from as early as 2143. In that year, he chose A.G. Robinson over Jonathan Archer to pilot the NX-Alpha. This decision was a difficult one for Forrest, since the engine of the NX-Alpha had been designed by Archer's father, Henry.
Robinson easily broke the Warp 2 barrier, but the ship exploded shortly after reaching warp 2.2. Although Robinson survived by jettisoning an escape pod while at warp, the Vulcans claimed that the warp program should be postponed. Forrest stopped at the 602 Club, a bar frequented by Starfleet personnel, and informed Archer and Robinson that the program would be suspended. After Archer and Robinson disobeyed orders and ran a new test that was successful, Forrest persuaded Starfleet Command to reinstate the program. (ENT: "First Flight")
The Enterprise mission
In 2150, the final selection for the captaincy of Enterprise NX-01 took place between Captains A.G. Robinson and Jonathan Archer. Although Captain Gardner had originally been considered the most suitable choice by Vulcan Ambassador Soval, Forrest – by then an admiral – assigned Jonathan Archer as the commanding officer of Enterprise in October that year. (ENT: "First Flight", "Shadows of P'Jem")
After Klingon courier Klaang crashed on Earth during the Broken Bow Incident in April 2151, the Vulcans wanted to delay the launch of Enterprise and, as the Klingons had requested, to return Klaang to his homeworld: Qo'noS. Suspecting this approach wouldn't sit too well with Archer, Forrest requested that he immediately attend a meeting at Starfleet Medical, which Forrest also attended, along with a couple of Starfleet officers and a few Vulcan dignitaries, including Ambassador Soval and Sub-Commander T'Pol. While Archer was yet to make an appearance, Forrest respectfully contended that Starfleet had a right to know what was happening with the case. Once Archer had arrived, the admiral helped explain the situation to him. Forrest initially warned Archer that Starfleet might have to defer to the Vulcans' judgment. However, Forrest listened to Archer argue otherwise, and, moments later, the admiral overruled the Vulcans and declared that Starfleet itself would return the Klingon. Forrest authorized Enterprise to launch early, on a mission to return Klaang to Qo'noS alive, despite protests from Soval and the other Vulcans. Forrest told the ambassador that Humanity had been waiting nearly a hundred years for this chance, and, after the Vulcans departed, he told Archer not to screw it up.
One condition of the launch, though, was that T'Pol join the ship's crew; Forrest advised Archer that he and the rest of the crew should think of her as essentially a chaperone. Forrest also officiated over the launching ceremony of Enterprise. At a press conference immediately prior to the mission launch, he said that Humans had been ankle deep in space travel and that it was time to swim. Forrest also cited, in his speech, the work of Doctor Zefram Cochrane and Henry Archer. Rather than quoting Cochrane, the admiral opted to play a video of a historically significant speech that Cochrane himself had given, at the dedication ceremony for the Warp Five Complex. (ENT: "Broken Bow") Forrest also stated, before the launch, that Enterprise would be making history with every light year, which Captain Archer later alluded to in his captain's log while pondering whether to venture aboard an apparently derelict Axanar cargo ship. (ENT: "Fight or Flight")
As his role was rarely to dispense information, he enjoyed notifying the Vulcan High Command about Suliban individuals whom Enterprise encountered during its initial mission. After the ship successfully completed that mission, Forrest related this enthusiasm to Archer and advised that he saw no reason for Enterprise to discontinue its exploration of space. (ENT: "Broken Bow")
Forrest felt similar joy upon receiving sensor scans of a rogue comet which Enterprise thereafter studied. He ordered the ship to investigate an automated distress signal from the ECS Fortunate, an Earth Cargo Service freighter that had been attacked by Nausicaan pirates, as the next closest ship was over three weeks away. (ENT: "Fortunate Son")
Forrest had to constantly walk a delicate balance between the goal of Human space exploration and Vulcan misgivings about it. After the discovery of P'Jem as a base for spy activity against the Andorians, it was destroyed in an attack by the Andorians. The Vulcan High Command called for an end of Enterprise's mission. Forrest refused, so the Vulcans called off joint Earth-Vulcan fleet operations and requested Sub-Commander T'Pol to transfer off Enterprise. The Vulcans eventually relented after T'Pol and Archer saved Sopek, a senior Vulcan officer. (ENT: "Shadows of P'Jem")
While Enterprise was visited by a group of Vulcans who did not embrace logic, Forrest contacted Archer and asked him to have one of the Vulcans, named Kov, call his father, who was dying. (ENT: "Fusion") He also ordered Enterprise to transport Vulcan Ambassador V'Lar from the Mazarite homeworld. She had information on the corruption of government officials and her life was in danger. (ENT: "Fallen Hero")
After an accident seemingly caused by Enterprise killed 3,600 Paraagans when tetrazine was ignited by the ship, Forrest informed Archer that, at the urging of the Vulcan High Command and Starfleet, Enterprise's mission was canceled. After Archer presented evidence that vindicated Enterprise, Forrest reinstated the mission. (ENT: "Shockwave", "Shockwave, Part II")
After a science team in the Arctic Circle lost contact with Starfleet, Admiral Forrest was informed and led a team to investigate. They rushed to the site, only to find it abandoned. He found that the scientists were missing, as was their transport. Admiral Forrest called in Enterprise to find the transport. This was Humanity's first contact with cybernetic beings known as the Borg, although the Humans never found out what their name was. (ENT: "Regeneration")
The Xindi mission
After an initial Xindi attack on Earth and once Archer met with a humanoid from the future who gave him details about the Xindi, Forrest met with Archer at Starfleet Headquarters. At first, Admiral Forrest was not convinced about Archer's information. Forrest was skeptical that it justified a mission into the Delphic Expanse. After Archer provided evidence by proving that the remains of a Xindi probe that had carried out the first attack on Earth included some debris which was from the future, Forrest authorized Enterprise to depart on a mission into the Delphic Expanse and to conduct a search for the Xindi there. (ENT: "The Expanse")
After the Xindi mission succeeded, Forrest welcomed the Enterprise crew home as heroes. Archer was debriefed by Starfleet Command and the Vulcan High Command. When Ambassador Soval insinuated that trellium-infected Vulcan ship Seleya was destroyed without the Enterprise officers doing enough to help the Vulcan crew, Archer became outraged, blaming the Vulcan High Command for not providing enough assistance in their mission. Admiral Forrest ordered Archer to take some time off to clear his mind. (ENT: "Home")
Forrest visited the United Earth Embassy on Vulcan in 2154. There, he had a conversation with Ambassador Soval, discussing joint missions between Earth and Vulcan. When a terrorist bomb detonated in the embassy, Forrest pushed Soval out of the blast wave, sacrificing himself. This proved to be Forrest's last act, as he was killed by the explosion. Captain Archer was saddened by the loss of his friend. Ambassador Soval, for his part, offered the captain his full support in the investigation and urged him to look on Vulcan for the answers he needed. (ENT: "The Forge")
In an alternate timeline, Forrest sent a message to Enterprise after meeting with the Command Council and gave Sub-Commander T'Pol a field commission to captain. Later, when Tucker accused T'Pol of allowing their mission to take "one wrong turn after another," T'Pol suggested that if he believed that she "acted improperly", she suggested that he contact Admiral Forrest. (ENT: "Twilight")
The character of Forrest was named after DeForest Kelley, the Star Trek: The Original Series actor who played Doctor McCoy. ("Broken Bow" text commentary, ENT Season 1 DVD/Blu-ray; ) Forrest's first name, "Maxwell", wasn't established onscreen until ENT Season 4 installment "The Forge", the only episode that uses that first name. The full name was included, however, in non-dialogue description in the pilot episode's schooting script.
While Rick Berman and Brannon Braga were developing Star Trek: Enterprise pilot episode "Broken Bow", they had no idea that Forrest would become a long-term recurring character. ("Broken Bow" audio commentary, ENT Season 1 DVD/Blu-ray) Hence, in a character breakdown sheet that Paramount sent to various talent agents upon casting "Broken Bow", Admiral Forrest was described as a non-recurring role, scheduled to appear only in the pilot episode. He was, in the same document, further characterized thus; "Human male. 50s to 60s. A career military man who's the highest ranking officer in Starfleet. He's fond of Captain Archer and has personally selected him to command Enterprise."  No reference to Forrest was made in the ENT series bible.
The process of casting Vaughn Armstrong as Admiral Forrest turned out to be relatively straight-forward. "When I went to audition for Enterprise, it seemed to go well," the actor noted. (Star Trek Magazine issue 118, p. 48) Although he auditioned (for Rick Berman) to play the part of Ambassador Soval, the creative staff who witnessed Armstrong's audition evidently had other plans in mind. ("Admiral Forrest Takes Center Stage", ENT Season 1 DVD and Blu-ray special features) "When I got home, I immediately got a phone call," the actor reminisced. "They said, 'As soon as you walked into the room, they saw Admiral Forrest!' I was thrilled to get a human on Star Trek, and praying that it would be a recurring character beyond that one episode." (Star Trek Magazine issue 118, p. 48) Armstrong was also extremely proud that the casting personnel for Star Trek: Enterprise had immediately recognized him as right for the part. 
Vaughn Armstrong was told nothing about Forrest's personal history. This allowed Armstrong the freedom to concoct his own backstory for the character, influenced by the scripts he was provided. 
Vaughn Armstrong found adopting the role of Forrest, for "Broken Bow", was somewhat odd, as he was much more used to playing aliens. Regarding the character's debut, he remarked, "It was great! Though I didn't have a lot of time to memorize my lines, because I usually do it in the makeup chair [....] This time I could actually sit in my trailer and learn my lines, but it was unusual in that there wasn't anybody patting my face while I was doing it!" (Star Trek: The Magazine Volume 2, Issue 9, p. 42) In fact, the make-up application time for Forrest was only ten minutes. 
For "Broken Bow", Vaughn Armstrong shot the scene in which Forrest, at Starfleet Medical, okays the launch of Enterprise on 21 May 2001 (with a make-up call at 6:15 am, a set call at 7:00 am, dismissal from the set at 8:05 pm, with fifteen minutes permitted for make-up removal). Armstrong filmed Forrest's speech at the actual launch ceremony on 5 June 2001 (with a make-up call at 9:45 am, a set call at 10:30 am, dismissal from the set at 4:30 pm, with fifteen minutes again permitted for make-up removal). ("Broken Bow" call sheets and production reports)
During the making of "Broken Bow", the question of whether Admiral Forrest would recur was brought up. "During the pilot, they said that it could be a recurring role," Armstrong recalled. "With the fact that they're staying closer to Earth [than previous Star Trek series], I had hopes that Forrest would return."  Following the initial appearance of Forrest in "Broken Bow", Armstrong suspected the character would indeed return. (Star Trek: The Magazine Volume 2, Issue 9, p. 47)
In the first draft script of "Fortunate Son", Forrest had more dialogue than he does in the final version of that episode, all of which was in conversation with Captain Archer. For instance, Forrest wryly mentioned that he'd been told a particular comet would be visible from Earth in approximately ten thousand years. He also said the ECS Fortunate was "pretty far out there" and asked Archer to "keep those 'postcards' coming" regarding scans from space which Enterprise had been relaying back to Starfleet Headquarters, on Earth.
Although Forrest appears in only one scene in the final edit of "Fortunate Son", he was originally scripted (in the first draft of the installment's teleplay) to feature in two of the episode's scenes, having an additional discussion with Archer, and appearing on a monitor in the captain's ready room. In that second scene, Forrest mentioned the Vulcan High Command hadn't been much help in regard to Nausicaan ships frequently attacking the Fortunate and other freighters of the Earth Cargo Service in a particular sector of space. The admiral had also helped file a complaint with Nausicaa about the piracy issue, but he wasn't expecting to hear from the Nausicaans anytime soon, because their government was, according to the Vulcans, in a constant mode of "transition." Though Archer suggested setting up regular patrols or escorts to protect the freighters, Forrest reminded Archer that Starfleet didn't have the resources to cover such a large sector. In response to Archer expressing his frustration about the situation, Forrest admitted he wasn't pleased with it either but reminded Archer he was out in space as an explorer, not a policeman. Sympathetically, the admiral instructed Archer to get the Fortunate repaired and sent on its way, Forrest regarding that course of action as the best one the captain could take, at least for the time being. When Archer agreed to that, Forrest told Archer he was doing good work and then ended the discussion.
In the first draft script of "Shadows of P'Jem", Forrest had slightly different dialogue than he does in the final version of that episode. For example, he was more confrontational with Ambassador Soval, in the episode's teaser. Whereas, in the finalized edit of the episode, he apologizes to Soval for the loss of P'Jem and respectfully states that the Vulcan High Command needs to take some responsibility for the facility's destruction, Forrest said neither of these things in the first draft teleplay. Instead, he directly regarded it as a given that the Andorians would destroy the monastery at P'Jem, asking Soval how he thought they would react when they discovered it was being used as a cover to spy on them. Similarly, although Forrest, in the final cut, says he can't agree with Soval claiming that Enterprise has destabilized an entire sector since its launch and the admiral then proceeds to remark that the situation was already volatile a long time prior to Starfleet getting involved, Forrest didn't say these statements in the first draft script either. Rather, he less diplomatically exclaimed, "If you'd told us what the hell was going on, maybe your sanctuary would still be there!" A remnant of that comment remains in the final version of the installment, as Forrest tells Soval, "Perhaps if you'd been a little more open with us, this tragedy might have been avoided." In the original draft of the teleplay, Forrest's part in the scene then continued much as it does in the episode, except for the admiral asking how long joint Human-Vulcan fleet operations would be suspended, to which Soval replied, "Indefinitely," before leaving the room. The two versions of a later scene, featuring a long-distance conversation between Forrest and Archer, were much more alike than the admiral's discussion with Soval. However, Forrest's transmission to Enterprise was originally referred to as a "priority" transmission, a status it is not given in the final edit. Also, during his discussion with Archer, the admiral commented, in the first draft script, that the Andorians were "a violent, predatory species," whereas he makes a comment about Andorian commandos in the final version, saying they "are a dangerous group."
By the time Forrest made his fourth appearance (in the first season episode "Fusion"), Vaughn Armstrong had thought up his own backstory for the character. "What I have picked out of the scripts indicates that I'm the highest-ranking Starfleet officer – I've been around Starfleet for a while. So I'm assuming that I worked with [Archer's] father," Armstrong reckoned. "I watched Archer come up through the Academy and enjoyed his abilities, his human-ness, his spontaneity and his intelligence. I sent him out there because I kind of look at him as being myself as a younger military man, and I really want to be out there myself, playing with the greatest toy in the universe. But they need me at home, so I send the favored son [....] I'm not sure that I'm not fond of Vulcans so much as I'm not fond of being told what to do, and Vulcans may not mean to, but they do that [....] I think part of the reason Forrest chose Archer was because he has a volatile nature but also real common sense. The common sense element is where we relate to each other." 
Meanwhile, Vaughn Armstrong also continued to consider what the future might hold in store for Admiral Forrest, such as hoping that he would soon need to make the decision of whether to reign Archer in and remind him what the reality of the situation was, Forrest being much more aware of a bigger picture than what Archer perceived of it. "I've got a whole fleet of ships out there," the actor said, "not as far as he is, but they're all dealing with people." Though he didn't expect to ever see Archer disobey Forrest, Armstrong hoped they would go through "confrontational moments" and that Forrest would be required to protect Earth from aliens. In fact, he imagined a feature film that would include Forrest being instrumental to the initial stages of a Klingon war begun on Earth. Armstrong also pitched the story for a time-travel episode which would have involved a Klingon attempt to prevent Forrest launching Enterprise. One thing the actor definitely didn't want to happen, during the run of the series, was for Forrest to be killed off. 
Vaughn Armstrong's familiarity with the Admiral Forrest character proceeded to develop over the seasons he played the role, though the character took more of a backseat in the third season of Star Trek: Enterprise, to make way for an overarching Xindi story arc in that season. (Star Trek Magazine issue 118, p. 47)
Forrest was referred to in a deleted scene from "E²". In that scene, Phlox recalled that Forrest had allowed Captain Archer to bring Porthos aboard Enterprise, Phlox citing this as potential evidence that Starfleet might allow children on-board starships someday. Archer then imagined "the look on the admiral's face when he sees two Enterprises pulling into spacedock." (ENT Season 3 DVD and Blu-ray special features)
By the advent of the fourth season, Vaughn Armstrong had devised more backstory for Forrest. "I pictured him starting in engineering, wanting to be the best engineer in town, and then he got involved with this warp engine of Archer's father," Armstrong speculated. "He got very excited about that, and then became a pilot, to hopefully one day fly the warp engine. Forrest was good at everything he did, so he came up through the ranks quickly. As time went on and he got more authority, he continued working with Archer's father and saw that his own job had become so important that he really couldn't fly the thing himself. He was probably a little older than he ought to be [to be a pilot], so he had to pick the best flyer to take this engine out." Armstrong went on to theorize that Forrest's decision to select Archer for the assignment was "because among other things, he thought the most human guy ought to represent Earth out there in the far reaches of space. So even though Forrest wants to be out there himself, there are too many responsibilities at home, so he had to pick the best man for the job." (Star Trek Magazine issue 118, p. 47)
The idea of killing Forrest off in "The Forge" was thought up by Manny Coto. (ENT: "The Forge" audio commentary, ENT Season 4 Blu-ray) Vaughn Armstrong believed it was important to attempt to retain a sense of surprise and shock for the audience as regards the character's death. (Star Trek Magazine issue 118, pp. 47 & 48) Ultimately, he felt that Forrest's demise was not as heroic as Damar's death in DS9: "What You Leave Behind". 
In the stage directions from the final draft script of "In a Mirror, Darkly", Forrest was described as "heroic" and "selfless".
In retrospect, Vaughn Armstrong concluded, "I wouldn't say that Forrest was my best contribution to the franchise."