Dr. Henry Archer was the primary developer of the first warp engine capable of reaching warp five, and was one of the principal designers of Enterprise NX-01. He was the father of Jonathan Archer and a good friend of Emory Erickson, the inventor of the transporter on Earth. (ENT: "Broken Bow", "Daedalus")
Henry worked closely with Dr. Zefram Cochrane to develop the warp five engine at the Warp Five Complex in Bozeman, Montana as part of the NX Program. Archer was also responsible for a derivation of the flux paradox more accurate than the one used by his Vulcan mentors. This breakthrough was the culmination of years of frustration with the Vulcans. (ENT: "Broken Bow", "Singularity")
Prior to Cochrane's disappearance in 2119, Henry took his son Jonathan on a tour of the Warp Five Complex, and introduced the boy to Cochrane, Tasaki, and other scientists working at the facility. Jonathan later revealed he did not know the significance of his father's work at the time. (TOS: "Metamorphosis"; ENT: "Singularity")
In 2121, Henry helped his son build and fly a remote-controlled starship model. During this time, the elder Archer admonished his son's disrespect for the Vulcans, whom Jonathan believed were responsible for holding back Humanity's advancement into space. Although Henry insisted that Vulcan Ambassador Soval (whom Jonathan referred to as "Ambassador Pointy") had been very helpful, he also expressed confusion and disappointment at the Vulcans' refusal to share their technology, lamenting, "They have their reasons. God knows what they are."
Henry assisted Jonathan in flying his model starship on a beach, but the youth found keeping the craft in the air to be difficult. When the toy crashed, Henry advised his son that he "can't be afraid of the wind. Learn to trust it." The younger Archer took this advice to heart thirty years later, when he made the decision to abandon his prejudices against Vulcans and start trusting Sub-Commander T'Pol. (ENT: "Broken Bow")
Henry spent the last two years of his life experiencing frequent pain and hallucinations, talking to people who weren't there, and often not even recognizing his son or wife due to advanced Clarke's Disease. He died in 2124 before his warp five engine could be constructed, since the Vulcans had held back the engine's development program. As a result, Jonathan Archer's resentment towards Vulcans grew into a grudge which held until the 2150s. (ENT: "Broken Bow", "Cold Station 12")
In 2152, a biography on the life of Henry Archer was in the works. Jonathan Archer was asked to write a one-page preface for his father's biography, a task he put off for weeks as he found it difficult to sum up his father's life in one page. He ultimately completed it with feedback from T'Pol. One draft of the preface included the following:
"How does one measure a man's legacy? Is it defined [by] the works he's created....the technological advances that will forever alter the course of Human history? If so, then no man since Zefram Cochrane himself has made a more lasting contribution to the future of Humankind than my father, Doctor Henry Archer. When I was about eight years old, my father took me on a tour of the Warp Five facility outside Bozeman, Montana. He introduced me to the people he worked with... scientists with names like Tasaki... and Cochrane. At the time, I didn't realize the significance of those names... or the significance of my father's work."
A further excerpt mentions "Soval's passive aggression" and that "the Vulcans would continue to obstruct Archer's work as they always had". They "never imagined that Archer's elegant mathematics might have been even more accurate than those of his mentors from Vulcan." It also says that "months — years — of frustration" prevented the "paradox of the Vulcans' ego-driven blindness" from being as funny as it might otherwise have been to him. (ENT: "Singularity")
Henry Archer was played by actor Mark Moses in his second of two Star Trek appearances. The shirt Moses wore in the final scene of "Broken Bow" was sold off on the It's A Wrap! sale and auction on eBay. 
As evidenced by the second draft of the script for ENT: "Broken Bow" (dated 1 November 2000) as well as by the episode's shooting schedule, this character's first name was originally to have been "William". The change from that to "Henry" was listed as one of six revisions of character names in a one-page "script note" at the start of the final draft script of ENT pilot "Broken Bow" (the page was dated 1 May 2001). The revised final draft script of "Broken Bow" (dated 11 May 2001) simply described Henry Archer, when he appears in the teaser of that installment, as "in his 40's [...] wearing 22nd century civilian clothing." When he is thereafter shown with Zefram Cochrane, the script pointed out, "[He] is only a few years younger than when we saw him in the opening scene." 
There appears to be some contradiction as to when Henry Archer died. "Cold Station 12" stated that he passed away when Jonathan Archer was twelve years old, or roughly 2124; dialogue in "Daedalus", however suggested that he was still alive when Jonathan entered flight school. Relating back to a reference stated in "Twilight", it was established that Jonathan graduated from flight school when he was twenty-four years old, or roughly 2136, suggesting Henry was still alive in the early 2130s.
Inconsequently, the latter timeframe was supported by a scripted but ultimately omitted scene from an early draft of "Home", where Jonathan was to have visited his father's grave on Earth. According to Henry's grave marker, he was born in 2077 and died in 2133. (ENT Season 4 Blu-ray special features)