HELLO,  SWEETIE!
 

River Song is Amy and Rory's daughter, also known as Melody. Also known as Mels. It's complicated.
River was born with the power to regenerate but used up all her remaining regenerations when she saved the Doctor after she poisoned him because she was programmed to kill him... Very complicated.
He first (more or less) met her on the planet of The Library, so her journal is (or will be!) packed with information the Doctor feels he shouldn't know about yet. The tragedy is, he knows how River dies, so when the two of them have that picnic at Asgard and do whatever they did together at the Singing Towers, he'll have a terrible secret that he won't be able to tell her.
River shares the Time Lord's buccaneering sense of adventure, as well as his knack for getting into trouble. But as for just how much they meant to each other: spoilers!

Melody Pond is the daughter of Rory Williams and Amy Pond. She was conceived on the TARDIS while it was in flight. Exposure to the Time Vortex as such a young a vulnerable stage altered her DNA the way it altered the DNA of The Doctor's people. These changes have given her abilities like The Doctor namely that she can regenerate.
Melody was kidnapped and brainwashed to kill The Doctor by Madame Kovarian (who worked for The Silence). She was later revealed to be Mels, Amy and Rory's childhood friend.
She regenerated into the woman known as 'River Song'. In the forest where she was raised for a time, they have no word for Pond. "In the forest the only water is the river". That mantra led to Melody Pond changing her name to River Song in order to hide her identity.
River Song is a sometimes companion to the Doctor. The River and the Doctor are romantically linked and eventually become married. Because the Doctor travels thru time and River usually does not (without him) they often meet in a non-chronological order. Generally speaking as we see the Doctor's relationship with River grow River's knowledge of the Doctor decreases. She is played by Alex Kingston.
River Song spends the majority of her time in a prison for the murder of "The best man she ever knew". That man is revealed to be the Doctor. That death however was not permanent and River frequently escapes the prison as she wishes.
As a child, Melody at some point left the care of Madame Kovarian and became indoctrinated by the Silence. She was made into a weapon to kill the Doctor. At an unknown point in her time-stream, while still only a little girl, Melody ended up in an orphanage in America in 1969 (where photographs were left of her and Amy) and later at a warehouse base in Florida. They put her into an Apollo astronaut suit that had been upgraded with alien life support systems. She used the suit to make calls directly to President Nixon in the Oval Office, begging for help. This led to her first physical meeting with the Doctor (in their meeting as a baby she had been a Ganger), though a brief one, due to her escaping from the spacesuit.
She moved to England under unknown circumstances, where she searched for and befriended the younger versions of her parents, Amelia Pond and Rory Williams. Nicknamed "Mels", she became close friends with them and helped them realise their love for each other. During this period in her time stream it is unknown where, or who she lived with; however it is highly possible that Madame Kovarian, or The Silence arranged for her to be raised by adoptive parents.

Also known as:	Melody Pond / Mels
Home Planet:	Earth (More or less)
Random ‘Yes!’ Moment:	Confronting the Stone Dalek.
Fact!	She once tried to kill the Doctor using poisoned lipstick!
First Appearance:	Silence in the Library
Most Recent Appearance:	The Wedding of River Song