Note that the names differ slightly in the English PSX version than in the Japanese version.

Bartz
Bartz is a wandering explorer, wh travels only with his pet Chocobo, Boko. He is heading towards the kingdom of Tycoon when a meteor blast strikes nearby and he finds two injured people near it.

Reina
Princess of the King of Tycoon, her father left for the wind shrine when he sensed that the crystals were in danger. For Reina, the adventure begins when she sets off to find her father.

Galuf
Galuf was nearby when the meteor hit, the impact of it caused him to develop amnesia. It takes Galuf the first half of the game or so, to remember the important quest we was on.

Faris
A mysterious character, who like Galuf has several questions about his past. Faris is the leader of a group of surprisingly loyal pirates and the master of Hydra, a gigantic sea serpent.

Krile
Where as you will have Reina, Galuf, and Faris join your team quite early on in the game you will have to wait until the later part to meet up with Krile. If I said much about her I'd spoil the game.