The Avatar system can be accessed from the Avatar Manager button in the My Iczer page, and the Avatar section of the shop. There are nine categories of Avatar. Main body, Hair, Equipment, Armor, Background, Moral Support, Accessories, Backpack and Potions. There are also some Special Avatars that allow you to have something a bit more unique.
All avatars can be purchased with either Gran, Fight Money, both, or one or the other. Look below the Avatar to see what payment options are available. Before you buy anything, hit the "Preview" button to see what it looks like on your avatar. This makes sure it looks right. There are lots of strange and unusual avatars around, and not all of them look right with all of the equipment or backgrounds. The Avatar system may also stop you from using equipment not designed for your pose.
Partners are Characters from the game and story who stand by you and talk to you as you play (though text), giving simple advice, cheering you on, and sometimes taunting your enemy. One could say that Partners are similar to the Paper Clip from Word, but my answer to that is that is that the Paper Clip isn't sexy, isn't a maid, and doesn't call you "Master." Enough said.