The Crystarium is a character’s growth chart, with which you can buy various upgrade nodes:

• Ability node teaches character a new ability.
• HP node increases character’s maximum HP.
• Strength node increases character’s Strength stat.
• Magic node increases character’s Magic stat.
• Accessory node increases character’s accessory slots by 1.
• ATB node increases number of segments in chraracter’s ATB gauge by 1.
• Role node raises character’s role level by 1.

You can purchase nodes by expending CP, which are obtained by winning battles. The characters’ growth is fairly linear, with the odd path branching off here and there. Note that some abilities are not available to all characters - for example, Vanille is the only one who gets access to the Death spell. See the Crystarium section of the guide for a detailed look at an optimal progression through the game for each character.