(covers information from several alternate timelines)
Kamikaze was a word of Japanese origin that translated to "Divine Wind". The kamikaze became famous in World War II, where Allied forces used the term to refer to Japanese pilots who made suicide runs, flying planes loaded with explosives into Allied ships. It was sometimes employed by starships, usually as an attack of desperation.
In an alternate reality created by Nero's arrival in 2233, George Kirk, acting captain of the USS Kelvin, made a kamikaze attack on Nero's ship, the Narada, once the crew were safely away in shuttles. (Star Trek)
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 2367, Captain William T. Riker was prepared to ram the USS 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 kamikaze attacks against invading Klingon forces, destroying or crippling fifteen ships. (DS9: "Tears of the Prophets")
Also in 2374, in an alternate timeline, the USS Voyager performed a kamikaze move against the Krenim weapon ship, causing the destruction of both vessels. The collision caused catastrophic damage to the ship's temporal core and instigated a temporal incursion upon the core itself, and the ship and all associated events were erased from history. (VOY: "Year of Hell, Part II")
In 2379, during the Battle of 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 fly their planes into explosive crashes.