Tl;dr: we don't know yet.
Long version: She introduced herself as "Missy" and other than that, the only thing we know about her is that she hints at a romantic relationship with The Doctor.
Given the line earlier in the episode where The Doctor says that "Somewhere out there, there's a woman interested in keeping us together", it's implied that not only did she give Clara the phone number to the TARDIS, she also placed the ad in the newspaper about The Impossible Girl.
As for her identity, to me it makes sense that she's some incarnation of River Song in whatever virtual reality Ten uploaded her into; after all, the half faced man is a robot (albeit a very mechanical one) and a virtual reality is a little like robot heaven.
I've also seen theories that she's the TARDIS incarnate again, a la "The Doctor's Wife".