There's this bar with moves on the bottom of the screen. The one in the middle recharges by letting the character perform normal attacks (this happens automatically unless you order him/her to use another attack). That's where Shulk keeps his Monado skills while other characters have their own special attacks. For instance, Dunbans special consists of three attacks you need to time with the b-button. I'll let someone else fill you in on his attack phrase, because I don't want to hog all the fun.
Next to the special attack are a bunch of skills that all recharge at their own pace. Each character learns a whole bunch of them and you need to come up with a proper movesets yourself. Skills can be improved as you progress and gain more experience.
Sometimes a party member will need some help or cheering up. That's what the b button is for. Using this will also improve the relationship between characters.
As you fight, the party gauge will fill up. You can use it to revive fallen allies or, if it's completely full, stop time and unleash a combo attack in which all active members can participate. The better the characters get along, the longer the combo can become. When you follow up a skill with a skill of the same colour, that attack will be much stronger. These combo's can be used to knock down and stun huge enemies or do massive damage.
Sometimes Shulk has a vision. You can approach an ally and give him/her an order to keep the bad future from happening. Time will stop while you give the command and it will cost one bar from the party gauge.
You only control one characters, the others are controled by AI. Aside from one or two characters, the AI is pretty competent. There are no gambits or whatever. You change their skillsets instead.