Professor River Song is a human archaeologist.
She is assumed to have an unusually intimate relationship with the Doctor, since she is the only person to know his real name. Unlike most classic series companions but like most new series companions, she seems to take shorter trips with the Doctor, living a more or less normal life between trips, rather than have one extended trip with the Doctor and never see him again after parting ways.
Biography
Professor River Song was an archaeologist from the 51st century who is already highly familiar with the Doctor when they first meet (from his perspective). She is, in fact, a future companion of his; she travels with him off and on, and he trusts her implicitly. When her team ran into trouble during an expedition to a deserted library, it was only natural that River should try to summon the Doctor by sending him a message on his psychic paper. But the Doctor who turns up, though recognizably River's Doctor, has not yet met her. Though bewildered, he agrees to help River and her team.
River's expedition to the planet-sized Library was financed by Felman Lux and was chartered to find out what disaster had happened there, 100 years previously, that had caused 4022 people to disappear.
River and the Doctor appeared to be very familiar, even considering the strong bonds the Doctor forms with his companions. She keeps a worn and battered diary whose cover looks very much like the TARDIS doors--apparently a journal of her travels with the Doctor. She also carries an upgraded version (in addition to the usual settings, it has "red settings" and "damper settings") of the Doctor's sonic screwdriver--a gift from the Doctor, she claims, even though he insists that he "doesn't give anyone" his sonic screwdrivers. (She responds, "I'm not 'anyone'!" and refuses to elaborate.)
As the situation in the Library deteriorates, River realizes she must prove to the Doctor that she is (or will become) someone he trusts completely. With profuse apologies, she whispers something into his ear... his true name. The Doctor is stunned--the name that neither the Carrionites nor the Pyrovile could discover, the name that Madame de Pompadour saw was "more than just a secret", his future self will freely divulge to River Song.
River died when she chooses to sacrifice herself in order to stop the Doctor from sacrificing himself in order to save Donna and the other 4022 people trapped on the Library. As he looks on, unable to stop her, she reveals that he must have always known how she was going to die. She pleaded with him to not try and change history, not to change one moment of what is to come for them.
After saving all the trapped people, the Doctor realizes his future self must have given her his sonic screwdriver for a reason. Inside he finds a data chip, which contains River's Data Ghost. The Doctor rushes to save her by uploading her into the virtual world contained in The Library's data core. CAL, the library's data core controlled by a human girl wired into its mainframe, also managed to save the data ghosts of all of River's dead archaeological team, so she would have company in the virtual world.
As the Doctor leaves the Library, he snaps his fingers to open and close the doors of the TARDIS, smiling when they comply, looking forward to meeting River again. (DW: Silence in the Library/Forest of the Dead)
When Davros said to the Doctor "Everywhere you go there are always deaths" River appears in a flashback, as the Doctor has flashbacks over those who have died during his travels. (DW: Journeys End)
It has been rumoured that River Song is either the Doctor's future wife, or it has even been suggested that she is a mutation of the TARDIS
Trivia
River Song is a reference to Mark Twain's Huckleberry Finn.
At Comic-Con 2008, Steven Moffat let slip that River Song has met other incarnations of the Doctor.