I've never had one but everyone who has had one told me they were really disappointing with them (which is why I never got one). From what I've heard from GMs and people who have them, what they do is they talk to you like before and during matches, and over the course of the match they'll say little obvious things like suggesting you revive your closed units or consoling you when you lose LP. Some people have compared partners to that annoying little paper clip in microsoft office that no one seems to like. They don't provide any actual tactical advantage, they basically just tell you what you already know and maybe amuse you with the things they say (though most people I talked to who had them were not amused, just annoyed). Which partner you get may effect things as well since most people I talked to bought the cheap ones.
The bunnies of Lavato have special abilities, like 'Action Skill: Make Carrot Disappear.'