A "pair" is just something fun to do on the forum. It's a way to signify in your signature a close relationship you have with another forum member. For some people, their pair is a good friend. For other people, their pair is someone that they are romantically involved with. So all in all, a "pair" on PC is a term given to a strong relationship between two people who agree to call their relationship by that term.
Also, goodness your avatar is adorable.