Konan was a member of Pein's team along with Yahiko back in the rain village during the civil war(s), there team was taught by Jiraya for a period of time. Konan's parents were killed at an unknown time prior to this event.
Konan's jutsu/abilities feature a large amount or origami (the art of paper folding). Konan can actually turn herself into origami and travel to the Rain village she helps command with Pein. Her weakness obviously is water because paper sticks together when it is wet, this is how she is easily defeated by her old sensei Jiraya when they meet again.
Konan also is the only member of the Akatsuki that calls Pein by his name, and also knows that Tobi is actually the leader of the group and not Pein, having been with him when Tobi was speaking to him. The rest of Pein and Konan's relation is unknown, but they appear to be only team mates.