That's the thing, we don't know... yet?
All we have at the moment are implications about their relationship. It's assumed, because of the fact she knows his real name (a fact not even avid watchers of the show know) assumes they were unusually intimate (even more than with Rose and, possibly, Martha); this seems to be reinforced by the fact that the Doctor gives her an improved Sonic Screwdriver, containing a Neural Relay (her "digital essence" in a way), plus (possibly) their picnic at Asgard.
However, as was said before, these are only suppositions, and the true nature of their relationship shall only really be revealed later on, possibly right near the end of Doctor Who's run (provided they don't decide to rebuild the Time Lord society and/or have the Doctor's regenerations "rebooted").
Also theres the fact that he said that there was only on time he would or could tell anyone his real name.