(covers information from several alternate timelines)
"They're showing us how far they're willing to go."
A suicide run, direct run, or simply ramming, was an attempt to collide one object with another as a means of inflicting maximum damage; this was sometimes employed as an attack of desperation. Such an act coincided with engaging a starship for ramming speed, while set on a collision course with its intended target.
In an alternate reality created by Nero's arrival in 2233, George Kirk, acting captain of the USS Kelvin, made a suicide run on Nero's ship, the Narada, once the crew were safely away in shuttles. The attack disabled the Narada and allowed the shuttles to escape. (Star Trek)
In 2256, Captain Philippa Georgiou intended to pilot a worker bee carrying a photon torpedo warhead into the Sarcophagus. The tactic became unnecessary after another means of delivering the warhead availed itself. In the moments to this decision, Georgiou had just witnessed a cloaked Klingon cleave ship plow into the USS Europa, which, in turn, initiated an auto-destruct protocol and destroyed both ships. (DIS: "Battle at the Binary Stars")
In 2258 of the alternate reality, Spock, having taken control of the Jellyfish, lures the Narada away from Earth, then proceeds to ram the craft into the hulking ship, igniting the red matter inside. Spock was transported away before impact. (Star Trek)
In 2259 of the alternate reality, Khan Noonien Singh, in grief after assuming the other Augments were killed when Spock had the advanced long-range torpedos detonated within the USS Vengeance, ordered the crippled starship to Starfleet Headquarters with the intent on murdering as many people as possible in revenge. The attack saw numerous buildings, including Alcatraz prison, destroyed and thousands killed. (Star Trek Into Darkness)
In 2267, Commodore Matt Decker piloted a Class F shuttlecraft into the mouth of the planet killer, attempting to destroy it after it had killed the crew of his ship, the USS Constellation. Decker was killed, but the explosion briefly caused damage to the interior of the machine, allowing James T. Kirk to devise a way of disabling it with a larger explosion. (TOS: "The Doomsday Machine")
In 2366, the Angosian prisoner Roga Danar made what seemed to be a suicide run when attempting to elude the pursuit of the USS Enterprise-D, but turned out instead to be making a tactical maneuver to force the activation of the vessel's deflector shield, thus disengaging the tractor beam. (TNG: "The Hunted")
In 2367, Captain William T. Riker was prepared to ram the Enterprise-D into a Borg cube that was attempting to assimilate Earth, but Lieutenant Commander Data was able to take control of the Borg's regeneration cycle and cause the cube to self-destruct. (TNG: "The Best of Both Worlds, Part II")
In 2370, a Jem'Hadar attack ship collided with the USS Odyssey's deflector dish, causing its destruction. Despite the fact that the Odyssey was withdrawing from the battle, the Jem'Hadar made the suicide run to prove to the Federation how far they were willing to go to prevent others from interfering with the Dominion. (DS9: "The Jem'Hadar")
In 2373, at the Battle of Sector 001, Lieutenant Commander Worf intended to ram the USS Defiant into the Borg cube after taking severe damage. The timely arrival of the USS Enterprise-E prevented him from doing so. (Star Trek: First Contact)
In 2374, during the first Battle of Chin'toka, the severely outnumbered Jem'Hadar defenders employed suicide runs against invading Klingon forces, destroying or crippling fifteen ships. (DS9: "Tears of the Prophets")
Also in 2374, in a succesful effort to protect the modified Nanoprobes in develoment aboard the USS Voyager, a Borg cube rammed a Species 8472 bio-ship while travelling at warp. Despite the large difference in mass between the Cube and the bio-ship, both vessels were destroyed. (VOY: "Scorpion, Part II")
Also in late-2374 of an alternate timeline, the USS Voyager performed a suicide run against the Krenim weapon ship, causing the destruction of her vessel. The collision caused catastrophic damage to the ship's temporal core and instigated a temporal incursion upon the weapon ship itself, and the ship and all associated events were erased from history. (VOY: "Year of Hell, Part II")
In 2379, during the Battle in the Bassen Rift, Captain Picard ordered the USS Enterprise-E into a collision course with Shinzon's Scimitar after conventional weapons were exhausted. Once the vessels were entangled, he intended to destroy both with the Enterprise's autodestruct. However, the autodestruct system was damaged in the collision and the Scimitar was able to pull itself free. (Star Trek Nemesis)
In the novel "A Rock and a Hard Place" by Peter David, Picard is outraged by the actions of a maverick officer temporarily filling in for Will Riker; when Deanna Troi tells Picard that, as far as she can sense, the officer is not mentally unbalanced, Picard describes kamikaze pilots in Earth's history, saying that the pilots somehow entered a state of mind where they felt it was quite rational to load their planes with explosives and fly them on suicidal dives into enemy ships.
In the Star Trek Online mission "The Doomsday Machine", K'Valk, a Klingon soldier seeking to stop the mad plans of Ambassador B'vat, performs the same action Commodore Decker did in "The Doomsday Machine", taking a shuttlecraft and flying it into the mouth of the weapon, claiming his reasoning for doing so is to regain his lost honor after failing to take command of the IKS Targ. The act leads to the player using special Harg'peng Torpedos to destroy the weapon.