The main leveling system in the game is called the Crystarium System which closely resembles the Sphere Grid in Final Fantasy X. Instead of gaining experience points after each battle, characters gain CP (Crystal Points) which can be used to unlock abilities and increase character attributes. Every character has a Crystarium, and within it, different sections corresponding to each Role the character has available.
The Crystarium is not available at the beginning of the game but becomes available through story sequences. After that point, levels in the Crystarium become unlocked as the story progresses, essentially capping the amount of growth that can be achieved in the middle of the game. Every character initially specializes in up to three Roles, and the other Roles are not even available in their Crystariums until much later in the game.
Final Fantasy XIII brings back summoned creatures, Eidolons. The Eidolons include: Odin, Shiva, Alexander, and Bahamut, and newcomers Hecatoncheir and Brynhildr. When summoned, the Eidolon stay in battle while the characters accompanying the summoner leaves the party. There is also a new feature called "Gestalt Mode", which joins the summoned and its summoner.
Here are some of the basic step-by-step guide to character advancement through the Crystarium.
1.) Choose Your Role
In the Crystarium, each role has its own path of development. Select a role.
2.) Follow the Path
Hold down the X button to advance along the current path of crystogenesis. As you reach and obatain new crystals, your character will learn new abilities and increase attributes accordingly.
3.) Reap the Benefits
Abilities acquired through crystogenesis are role-specific. For example, the Ruin ability obtained as a Commando cannot be used when that character is serving as a Saboteur. On the other hand, attribute bonuses are universal and apply across all roles.
4.) Direct the Course
Advance far enough down a given path, and you will eventually come to branching points. Here, you must direct the course of crystogenesis.
5.) Explore the Possibilities
Over the course of your adventure, all roles will become available to all characters. Giving you the freedom to customize your party as you see fit. Be aware that at least one ability must be obtained before a newly acquired role can be used in battle.