We first saw "Oswin Oswald" in the Asylum of The Daleks
(Enemy: Daleks), The Doctor couldn't see her face but did hear her voice via the comms. She was very clever with computer stuff and died.
Next we see her in The Snowmen
(Enemy: The Great Intelligence), where her name is "Clara Oswin Oswald", but The Doctor doesn't see her full name till the end of the episode. The name on her gravestone + the Run You Clever Boy And Remember lights a bulb in his head and he realises that it was the same person in The Asylum of The Daleks. The same person died twice in two different times.
Next we see her in The Bells of St. John (
Enemy: The Great Intelligence) where her name is "Clara Oswald". She is pretty clever here but very bad with computers. She gets sucked into the data core via the WiFi by The GI. The Doctor saves her and she comes back but with her she brought something more - computer skills for one, who knows what else she got from there. The GI of course doesn't approve of that and tries to get her back into the data core. But she uses her newly gained computer skills to hack into their servers to find their location - while doing that she makes a new username for herself. Clara Oswald for the win. Oswin.
The Doctor spends rest of the season traveling with her and trying to find out who she really is.
In The Name Of The Doctor (
Enemy: The Great Intelligence) The Doctor enters his grave where his whole life is open and and preserved, accessible to anyone who knows the answer to The Question. "The Question that must never, ever be answered."
I would like to note here: The Silence knew this was going to happen. if you remember - The Silence was not a species, it was a religion and they had a prophecy.
"On the fields of Trenzalore, at the fall of the eleventh, when no living creature can speak falsely or fail to answer, a Question will be asked, a question that must never, ever be answered."
So technically they were trying to stop this from happening from the beginning - they were not the bad guys. They preferred a universe without The Doctor than a universe ruled by The GI or The Daleks or any of the other idiots.
Anyway back to the plot - the prophecy said "when no living creature can speak falsely or fail to answer" - River (Song) wasn't living, she was connected to Clara via a telepathic link - and she knew the answer to The Question so she said it and the grave opened.
The GI jumped into The Doctor's timeline where he had full access to his past and future and started to destroy his past - imagine The Doctor losing a battle against a villain who appears out of nowhere 1000 times.
To save The Doctor from this massacre Clara jumps into the timeline as well where she stops the GI every single time. Sometimes she is by herself and sometimes she meets The Doctor in his adventures (Asylum of The Daleks, The Snowmen) - she never quite remembers who she was - which is why Run You Clever Boy And Remember (Me).
At the end of the episode she is shown helping every single Doctor that has existed so far (including the Old Series) - Including the first doctor. Then 11th Doctor himself jumps into his own timeline to save Clara (causing it to collapse on itself). He finds Clara in this mess but Clara found someone - someone who is The Doctor but isn't.
Clara: Who's that?
The Doctor: Never mind. Let's get back.
Clara: Who is he?
The Doctor: He's me. There's only me here, that's the point. Now let's get back.
Clara: But I never saw that one. I saw all of you. Eleven faces. All of them you.
The Doctor: I said he was me. I never said he was the Doctor.
Clara: But I don't understand.
The Doctor: My name, my real name, that is not the point. The name I chose is the Doctor. The name you choose, it's like, it's like a promise you make. He's the one who broke the promise. {Clara collapses} Clara? Clara! He is my secret.
The Old Man: What I did, I did without choice.
The Doctor: I know.
The Old Man: In the name of peace and sanity.
The Doctor: But not in the name of the Doctor.
Continues on 23rd November.