Final Fantasy XIII wont feature the traditional Levelling system, it instead has the Crystalium System.

Weekly Shonen Jump has the answers for what this is in the latest issue. It's basicly used to give your characters new abilities and skills.

Every enemy you defeat in battle earns you some Crystal Points. These are used to purchase all the different skills and abilities available for your character. Upgradable skills range from stuff like your health to different magic types.

It is also thought that each character will upgrade per battle role, like Blaster, Attacker, Defender etc. When you upgrade Blaster, you gain Fire magic. This upgrades to other powerful Fire magic the more CP's you use on it.

Source - Andriasang