Who is Professor River Song to Doctor Who?

River Song is a woman from the Doctor's timeline, but they keep meeting in the wrong order. It is argued whether they are supposedly married in the future, but she carries around a blue book she claims to be her diary of all the times she has met the Doctor, but he hasn't met her at that point in his timeline yet.

In the episode 'Silence in the Library' River is an Archeolagist but in a more recent episode in series 5, 'the time of angels' she is younger and doesn't know of this future yet, meaning the Doctor lied when she asked him if he knew who she was as in the episode 'Silence in the library' River sacrafices her life instead of the Doctor, but the Doctor manages to save her to a computer so she can carry life for eternity in a virtual world along with her friends who were victims of the vashta nerada.

Hope this answers your questions.

Sorry just an add on. The Doctor didn't lie when he said he didn't know her in silence in the library. As you say, they keep meeting in the wrong order, Silence in the Library was the first time He has ever seen her, but she has had hundreds of adventures with him in her own personal past. We follow the doctors story so we and he had never met her.

Hope this helps you answer your question:

Basically the doctor is a time traveller so he travels in different times. For example the 21st century or the 5th century or the 51st century etc. so basically he meets Professor River Song at different times. That's why sometimes he knows her and sometimes he doesn't. But Professor River Song has already lived her life and the doctor is going to the past and present.