According to the glossary a partner is:
A person from Lavato who travels with an Iczer, giving them advice and learning from their duels.
You don't start with a partner, but they are sold in the shop (most for incredibly high FM costs, or the cost of 2 packs in Gran). It appears you can only have 1 partner at a time and each partner has a limited number of "uses". However the shop does not appear to explain what those "uses" do, just shows how many of them each partner has and what the partner looks like.
Also there is a weird partner who is free with has japanese text for its name and description and has a picture of a notebook computer. I purchased and equipped it on the alpha servers. Its effect seemed non existent however. At the beginning of a match it would show up on the screen and the word "null" would come out of it, suggesting to me that if it was translated/not glitched it would have said some sort of advice. However after the match I checked and the partner screen still showed it as having 20/20 uses so it seems that advice bit doesn't count as a "use".
The bunnies of Lavato have special abilities, like 'Action Skill: Make Carrot Disappear.'