[Wow, he's actually getting to do a battle tutorial! Neat.]

Wait mode allows the action to be paused while the Puppeteer is making their decisions in the menu. After all, it can be rather frustrating to attempt to access a healing spell, only to have your healer cut down in the same breath. As for Auto, it allows the Batter and his Add-On to make their own decisions, to whatever end that entails.

[Good if you're over-leveled and don't really want to bother.]