Some of this has been answered, but I'm going to launch into an explanation anyway just for kicks and cookies.

When you press circle, it brings up the menu and (on Wait mode) freezes everything. You then input all the actions you want, and when you press X the menu will go away and a bar, much like the ATB bar, begins to fill up. When it's full, the action is carried out on the target selected. Magic takes longer to fill than, say, Attack.

But things would get monotonous if you had to constantly freeze the action and manually tell everybody what to do because of the way the game is set-up. So you can use the Gambit system which let's you set your own A.I. One example is "If HP is < 30%" then "Cast Curaga". With this Gambit a character will automatically start to heal anybody whose HP is 30% or lower. Or "Cast Oil on enemy" followed by "Cast Fire on all enemies with Oil status effect" (oil increases fire damage). This will cause them to Oil an enemy then cast Fire until they die and then repeat the process on the next enemy without you telling them to do it. You can even set it up so they stop once their MP gets too low. MP also refills as you run around slowly, and some skills let you refill it faster (like refill so much everytime you hit somebody that sort of thing).

In other words, it's the same exact thing as always at its core. You select things out of a menu, and then you see the action carried out while the enemy and you alternate turns. It's still a turn-based RPG, despite looking more like an action-RPG, you just don't go into a battle screen.

The Gambits are optional and are only there for convenience. If you want to input everything manually then you can. You can turn them off or on at any time.

So it's really not like Vagrant Story =/

And that's about that!