HOW BADGES WORK: You may take a maximum of 9 into battle (3 badges for each fighter, not counting stolen badges)
You can equip badges whenever it's your turn. Some badges have an "equipping fee", which means you have to pay BP to put them on.
There are two kinds of badges: Passive badges, and Move badges.
Passive badges are badges that effect you during your turn. Ex: Defend Plus gives you +1 DEF at all times. At the start of each turn, you will have to pay some BP. Putting on multiple passive badges at once will increase the BP amount.
Move badges are badges that you have to use. Essentially, they're just extra attacks that everyone can use. Tactics on crack, if you will. They cost BP every time you use them, however.
If you don't have enough BP to use a badge, you have to use Refresh or some other BP restoring move in order to use them again.