A lot of important backstory comes from the first book. AGoT tells us why Robert Baratheon is so adamant about trying to kill Daenerys and Viserys and thus wipe out the Targaryen name completely.
On top of that, the story of Robert's Rebellion (which predates AGoT by about 14 years, as I believe Catelyn mentions that Ned was at war when Robb Stark was born) is detailed greatly, as it tells how the story of the Mad King Aerys Targaryen, how the Baratheons became the rulers of Westeros, and how the Targaryens are all but extinct. Robert's Rebellion ties many of the major characters together and helps define them, as Jaime and Tywin Lannister, Robb Stark, Gregor Clegane, and Robert and Stannis Baratheon (Renly was only a few years old at this point) all had important roles in this story.
As with any adaptation, there are changes that have to be made, and details that have to be left out. Unfortunately, that meant that most of the series' "prologue" (for lack of a better term) in the form of Robert's Rebellion and other stories that predate the events of the show were left out. There are pictures from unaired footage (most likely from the original pilot) that show the deaths of Ned Stark's father and brother at the hand of Mad King Aerys. It is likely the original pilot would have gone into some depth on this, but where film is the art of showing, not telling, giving long flashbacks like the book does (thus taking us out of the main story) would detract from the story and confuse watchers. Keeping this in mind, the show's writers put in the most important details, and that is enough to keep the show-only folks on track.
In short: The books have a TON of content pertaining to backstory, especially the first book. However, in order to keep a season's adaptation in 10 hours, a lot of material will still be cut. Unfortunately, a lot of the cut material was backstory. If you're interested in backstory, check out the books. They're definitely worth a read.