Trident
The Trident is a sacred relic that gives it's wielder the ability to control the sea, or Atlantica. It is more of a spellcasting weapon then a melee weapon. King Triton is the current wielder of the Trident.
Powers
Its main power is shooting a large blast of yellow energy at a certain target. Other powers it has, examples have shown, are manipulating water and changing the present state of a being. In Kingdom Hearts II, King Triton uses it to write his name on a contract. When wielding the Trident, Ursula uses some thunder and blizzard-type attacks during the second boss battle against her, but it is not certain that all these attacks were the power of the trident.
Story
Kingdom Hearts
In Kingdom Hearts, Triton is still the wielder of the trident. He uses its power to destroy a few Screwdivers in one instance, and then to destroy a crystal trident, that is the key to Atlantica's keyhole. Ursula eventually steals it again and, like the movie, grows to a gigantic size. Sora and co. defeat her and then use the trident as a replacement for the trident crystal to seal the keyhole.
There are also markers along the walls in Atlantica that point to the Palace.
Kingdom Hearts: Chain of Memories
The story starts out with Sora, Donald, and Goofy meeting Ariel as she is swimming past them carrying the trident. She tells them to not tell anyone that they saw her. Sora and co. enter the palace and find that the Trident was stolen and that Ariel has gone missing.
Later, Sora and co. find out that Ariel stole the trident but only to help Flounder who has been lost in the surface world. She is going to give it to Ursula because Ursula said it is the only way to save Flounder. The all travel to Ursula's Lair and find that she kidnapped Flounder as a way to get the trident, which Ariel gives over to save Flounder.
In the end, Ariel reclaims the trident and Sora suggest she should say she found the trident and got it back all by herself. But she says she can't do that because it was her mistake which she made to help her friend. She has to tell the truth when she returns the trident.
Kingdom Hearts II
In Kingdom Hearts II, the trident only appears in scenes with Triton and the song-battle against Ursula, where she uses it as a weapon again. During which, Ursula uses to deflect Sora's attacks and to attack Sora, Ariel and the others. At the end of the song, Sora successfully knocks the trident out of Ursula's hands and lands in Eric's boat. Eric then uses it as a javelin to defeat Ursula once and for all.
Etmology
A trident is the royal scepter of Poseidon and Triton in Greek mythology.