Final Fantasy X does not use the level system that was so common with previous Final Fantasy games. In Final Fantasy X, there is a thing called the “sphere grid”. Each character has his or her own grid, and each character can (after quite some sphere-advancing), enter another character’s grid. That way, you can end up with Tidus as your Black Mage and Lulu as your White Mage (meaning every character can learn every ability on the sphere grid).
To advance on the sphere grid, your characters gain experience (like in previous Final Fantasy games), but they gain “sphere levels” instead of just “levels”. Each sphere level allows your character to advance said number of nodes on his or her sphere grid. For example, you beat a boss and Yuna gains two sphere levels from that one boss alone, and she already had one sphere level from before the fight. She can now advance to up to three nodes in her sphere grid.
Throughout the game, you will find various different types of spheres (power sphere, magic sphere, key sphere, etc.) which can be used to activate the nodes found within the sphere grid. Keep in mind, though, that different spheres activate different nodes.
Ability: Activates nodes that teach your characters abilities or magic.
Power: Activates nodes that increase strength, defense, or HP.
Magic: Activates nodes that increase magic, magic defense, or MP.
Speed: Activates nodes that increase agility, accuracy, or evasion.
Key: Opens up “locked” nodes that prevents your characters from advancing past a certain point on their grid. Use the corresponding Level key with the corresponding Level of node.
Those are just the most basic nodes; there are many more to be found within the environment of Final Fantasy X. Characters can go over nodes already activated by other characters.