There are many important characters in A Song of Ice and Fire. Unlike most book series, A Song of Ice and Fire focuses on a large number of characters, not just one or two. While every character is important, only a few have been focused on since the first book. Those are the most important characters.
House Stark: Descended from Brandon the Builder, who built the mighty castles of Winterfell and Storm's End and the Wall, House Stark claims dominion over the lands north of the Neck, the narrow, swampy isthmus connecting the two halves of Westeros. Their main fortress is the ancient castle of Winterfell, which is almost 10,000 years old.
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Brandon Stark: The second youngest member of House Stark and called Bran by his family, Bran was six at the start of the series. In the first few chapters, he was thrown from a tower by Jaime Lannister for witnessing an affair between him and the queen, Cersei Lannister, his sister. Bran survived but was crippled and could not remember anything about his fall. He later found that he possessed magical abilities and was taken to the far north to find someone to teach him about his power.
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Arya Stark: The third youngest member of House Stark and the youngest girl, Arya never liked being a lady. Her father, Eddard Stark, the Lord of Winterfell, took her the capital of Westeros, King's Landing, to civilize her. Eventually, he gave up and hired a sword master to teach her how to fight. After Eddard was killed by the Lannisters, she ran away and witnessed the War of the Five Kings firsthand as she tried to reach Winterfell. Instead of reaching Winterfell, she was forced to leave Westeros altogether, going to the city of Braavos in Essos where she meet a group of assassins that she joined.
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Sansa Stark: The second oldest member of House Stark, Sansa was everything Arya wasn't. She loved singing, sewing, and dresses. When Eddard Stark's old friend, King Robert Baratheon, came to Winterfell, she was betrothed to Joffrey Baratheon, the crown prince. However, when Robert died from a plot by his wife Cersei, Joffrey became king and had Eddard executed in front of Sansa. He then held Sansa captive and had her punished for being a Stark every day. Sansa was eventually set aside from marriage to Joffrey and was instead married to Tyrion Lannister. On Joffery's wedding day, he was poisoned and killed. In the mass confusion Sansa was rescued and taken to the Vale of Arryn, where she could hide from Cersei, who blamed her for her son's death.
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Jon Snow: The illegitimate son of Eddard Stark, Jon was raised at Winterfell, much to the chagrin of Catelyn Tully, Eddard's wife. At the start of the series, he decided to join the Night's Watch, an ancient brotherhood sworn to man the Wall, a gigantic wall of ice 600 feet tall, which separates civilized Westeros from the northern wilds, where barbarians and ice demons thrive. After the death of the Lord Commander, Jon was elected as Lord Commander by his fellow brothers of the Night's Watch, as he had been able to hold a section of the Wall from a giant army of barbarians with no more than thirty untrained boys. Recently, Jon was the target of an assassination attempt by those who thought he was ruining the Night's Watch. Whether he survived is unknown.
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House Lannister: Descended from Lann the Clever, a hero from Westeros's ancient past, House Lannister is the richest of the great houses, ruling the Westerlands from the impenetrable fortress of Casterly Rock. The Lannisters are known for their trickery, having fooled the previous king of Westeros into trusting them before killing him.
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Tyrion Lannister: The youngest child of Lord Tywin, Tyrion is a dwarf, with mismatched eyes and stunted legs. He is also the wisest and most clever of Tywin's children, as his disabilities prevented him from training as a knight in his childhood. In the first book, Tyrion was taken captive by Catelyn Tully, who thought he was behind the attempt to kill Bran. After he was found innocent, he was sent by his father to serve as the King's Hand, the king's most trusted adviser, to prevent Joffrey from mismanaging the kingdom. During this time, Tyrion was severely injured while protecting King's Landing from King Stannis Baratheon, the rightful heir of Robert, as Joffrey was born of incest between Tyrion's siblings. Soon afterward, Tywin took the position of King's Hand and forced Tyrion to marry Sansa Stark so the Lannisters could have control in the North. However, Tyrion was framed for murder of Joffrey and imprisoned. His brother Jaime, the only family member who truly liked him, freed him. During his escape, he killed his father for lying to him that Tyrion's one true love had simply been a gold digger and had not loved him. He then escaped to Essos, where he traveled the continent to find the true ruler of Westeros, Daenerys Targaryen, the Mother of Dragons.
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House Targaryen: Descended from Aegon the Conqueror, who unified all of Westeros, House Targaryen was the monarchy that controlled Westeros. However, its rule was overthrown by Robert Baratheon when he rebelled against King Aerys Targaryen, also known as the Mad King. Every member of House Targaryen was killed except Viserys and Daenerys, the youngest children of Aerys.
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Daenerys Targaryen: The last surviving member of House Targaryen, Daenerys is also one of the greatest conquerors since Aegon. Born in the ancient Targaryen island citadel of Dragonstone as one the greatest storms in recent memory ripped apart her father's fleet, she was the youngest child of Aerys. Taken to Essos by men still loyal to her family, she spent the first 13 years of her life on the run from Robert's spies with Viserys, who often antagonized her. On her thirteenth birthday, Viserys forced her to marry Khal Drogo, the leader of a khalasar, or nomadic city, with over 40,000 strong and able warriors, in exchange for Drogo helping him regain his father's crown. However, when Viserys threatened Daenerys in a drunken rage, Drogo killed him by crowning him with molten gold. Shortly afterwards, Drogo took a wound in battle. The wound festered, killing him. Blaming Daenerys for his death, his mighty khalasar fractured into numerous smaller ones. At his funeral, Drogo was cremated along with his worldly possessions, including three petrified dragon eggs given to Daenerys as a wedding gift. The heat of the fire somehow made the eggs hatch, bringing the first dragons in centuries into the world. She then traveled east, finding the great city of Qarth where she lived for a time, letting her dragons grow, until she was forced to leave after she angered the denizens by burning the warlocks of Qarth. She then traveled back west, finding the slaver cities of Astapor, Yunkai, and Meereen. Though disgusted by slavery, she bought a slave army at Astapor, only to use it to sack Astapor, defeat Yunkai, and take over Meereen, installing herself as its queen. However, the native Ghiscari did not like her rule and conspired to kill her, hiring mercenaries and assassins. Before they could kill her though, one of her dragons flew her far away.

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"A Wiki of Ice and Fire." A Wiki of Ice and Fire. Wikimedia, 09 March 2012. Web. 20 Mar 2012. <http://awoiaf.westeros.org/>.