Old Dude gives a good background summary. Here's some character information
A "quick" run down on the most important characters introduced in the first episode: Most on this list are main/important characters.
On the continent of Westeros, land of the "Seven Kingdoms" previously conquered and united under the rule of the Targaryens.
House Stark -- Rulers of Winterfell, Wardens of the North.
Lord Eddard Stark [Sean Bean] -- Noble who helped Robert Baratheon usurp the throne from the Targaryans. Lord of Winterfell. Baratheon travels to Winterfell to ask Eddard to become the "Hand of the King" after the death of the previous hand.
Lady Catelyn Stark (nee Tully) -- Married off to Stark to seal the alliance between house Tully and House Stark. Has grown to love her husband deeply.
Robb Stark: Eldest Stark son (15) and heir apparent to Eddard Stark
Sansa Stark: Eldest Stark daughter (13). Betrothed to Prince Joffrey Baratheon.
Brandon "Bran" Stark: Middle Stark son (10): Fantastic climber, which leads me to discover the incestuous relationship between Jaime and Cersei Lannister. Gets thrown off the a Winterfell tower by Jaime Lannister.
Arya Stark: Youngest Stark daughter (8-9). Tomboy (hates her needlework!) and general little pain.
Ricken Stark: Youngest Stark son (5)
Jon Snow: (16?) Bastard son of Eddard Stark. In Westeros bastards always take the same surname in each region. In the north the surname taken is "Snow". Wants to join the Night's Watch on the wall as his position as a bastard makes his life rather increasingly uncomfortable in Winterfell.
Theon Greyjoy (16?): Ward/Hostage of Eddard Stark. In Westeros it's common for children to be taken as ward/hostages to force compliance of treaties, etc. Heir of House Greyjoy. Close friend to Robb Stark.
The Lannister Children
Cersei Baratheon (nee Lannister): Queen consort of King Robert Baratheon. Involved in an incestuous relationship with her twin brother Jaime.
Jaime "The Kingslayer" Lannister: Member of the Kingsguard (an elite guard for the King). Slayed the last Targaryen king while a member of the guard. Involved in an incestuous relationship with twin sister Cersei.
Tyrion "The Imp" Lannister: Dwarf son of Tywin Lannister. Younger brother to Jaime and Cersei. Highly intelligent and sardonic. Despised by his father.
King Robert Baratheon: Usurper to the Iron Throne. King of the seven kingdoms of Westeros. Once a great fighter, now a reluctant ruler: occupies his time with whores, hunting and drink. Leaves the ruling of his kingdom to his "Hand". Was once betrothed to Eddard Stark's sister who was kidnapped and presumably murdered by the Prince Rhaegar Targaryen during the rebellion.
Prince Joffrey Baratheon (12): Eldest Baratheon son. Betrothed to Sansa Stark.
On the Eastern Content (The Free Cities) -- ancestral home of House Targaryen
The Surviving Targaryens: Survived the sacking of King's Landing in Westeros and the Targaryen purge. Fled to the free cities on the eastern content after the rebellion.
Viserys Targaryen (late teens). Cruel, unstable and has grand ambitions (but no real means) of retaking the Iron Throne. Sells his sister to Khal Drogo to form an alliance.
Danerys "Dany" Targaryen (early teens). Younger sister to Viscerys. Born during the Baratheon rebellion. Victim of mental, physical and sexual abuse by her brother. Sold as wife to the warlord Khal Drogo to purchase the services of his army.
Khal Drogo: One of the most powerful warlords in the east. Commands an army of over 10,000 horsemen.
Jorah Mormont: Exiled knight who enters the service of Viserys and Danerys at the wedding of Dany and Drogo.