Basically after each battle players are given Crystal Points, which can be spent across charts defining a character's "role" as you go.
It's said to resemble Final Fantasy X's Sphere system, which ditches the traditional staple of experience gathering and levels. CP is the new EXP if you will for FFXIII, but instead of your character gaining levels you attribute the earned CP to 'level up' roles instead.
As certain roles become more invested in they'll open up more advanced versions of themselves, but there is a "Crystarium Level" in place which caps progress. This is an overall character level, sort of, and prevents roles from upgrading too much too quickly. As the game goes on this is raised allowing you to further boost roles.
Whether this is better than the usual method of level grinding will reveal itself soon enough, this December in fact as Japan gets Final Fantasy XIII on PS3. We here in the western world have to wait till next year, when it arrives on Xbox 360 and PS3.
IGN has translated some of Famitsu's ravings, and they have a lot more.