Here's a basic rundown as well as I can explain it. I've included some minor spoilers that will have to do with the really early episodes and the exposition of the series, so if you want to avoid them your best bet is to check out Wikipedia's summary.
This is a character driven show set in the fictional land of Westeros
, which is divided by seven realms (provinces) all ruled by their own lords. The seven realms were first united under one king named Aerys Targaryen. The story takes place years after that king has been killed in war, and a new king named Robert Baratheon (played by Mark Addy) has usurped his throne.
We start off in the middle of King Robert's rule by finding out that the King's right hand man and chief advisor, the "Hand of the King", has been mysteriously killed. Robert realizes there was foul-play and begins to worry that there are corrupt people under his rule that want him dead. He goes to Eddard Stark (played by Sean Bean), lord of Winterfell and the realm in the north of Westeros. Eddard (also known as Ned) has been friends with Robert for ages, so Robert sees him as the only person he can trust. Robert makes him the new Hand of the King, basically putting him in charge of ending the corruption before it's too late.
The show follows Ned and his family. He has a wife, Catelyn, and five kids: Robb, Sansa, Arya, Brandon, and Rickon... all of which have significant storylines that are affected by Ned's new job as the Hand. Ned also has a bastard son named Jon Snow, whom he illicitly fathered with some girl that Ned did the dirty with years ago when the country was last at war. Jon Snow decides to join the Night's Watch at The Wall. The Wall is in the far north of the kingdom and is the boundary that separates Westeros from the danger that lies past it. The Night's Watch is pretty much the group of rejects that patrol The Wall and prevent any sort of attack or invasion from beyond. Almost everybody in the Night's Watch has either committed some sort of crime or wronged someone important in the country, and rather than die/rot in prison, they join the Watch instead. Jon joins because his lot in life as a bastard son isn't looking very pretty, and both he and Ned believe he has the potential to excel as a guard of the Watch.
The other family the show focuses on are the Lannisters, the primary antagonists of the series. There's Cersei, who is queen to Robert, though she loathes him. There's her brother Jaime, who first served under the old king before turning on him and killing him in battle, receiving the nickname "Kingslayer" and a shitload of distrust from King Robert. Then there's Tyrion, born a midget which makes him the black sheep of the family, who uses wit as his weapon rather than blade.
There's also another storyline going on across the sea away from Westeros, where the son of the old king Targaryen has been exiled with his sister. The son, Viserys, wants beyond anything else to take back what was usurped... the kingdom of Westeros and the iron throne that rules it.
tl;dr version - the show is about the different ruling families in the kingdom of Westeros that are all jockeying for position to win the crown and control the country.
If you need me to clarify anything, feel free to ask... I probably didn't explain this as well as I could have. There's a lot to follow with this show but it's really a lot less confusing than it first may seem.