The War of the Four Kings is a major military conflict that erupted in the wake of the death of King Robert Baratheon. His son Joffrey took the Iron Throne, but it was claimed that Joffrey was a bastard born of incest between Queen Cersei and her twin brother, Ser Jaime Lannister. On this basis, Robert's younger brother Stannis Baratheon also claimed the throne. Robert's youngest brother, Renly Baratheon, also claimed the throne, claiming that he would make a superior ruler to the ruthless Stannis.
The situation was complicated further when Robb Stark of Winterfell, leading a host south to free his father and to relieve a Lannister attack on the Riverlands, was declared the King in the North by his lords bannermen (and some of the river lords), and he declared the North to be a sovereign kingdom no longer subject to the rule of King's Landing.
The current factions are involved in the War of the Four Kings:
King Joffrey Baratheon claims the Iron Throne by right of inheritance from his father, the late Robert Baratheon. He holds the capital city of King's Landing and his claim is supported militarily by the powerful and well-equipped armies of House Lannister. However, the Lannister armies are currently bogged down in fighting the Stark forces in the Riverlands, leaving the capital potentially vulnerable to assault by sea or from the south.
King Renly Baratheon, the youngest brother of King Robert, claims the Iron Throne by right of strength and conquest. He is supported by the armies of House Tyrell, who can field the largest armies in all of Westeros. With the Lannisters and Starks fighting one another north of the Blackwater Rush, there appears to be nothing stopping the Tyrell forces marching straight on King's Landing and installing Renly on the Iron Throne.
King Stannis Baratheon, the younger brother of King Robert and the elder brother of Renly, claims the Iron Throne by right of inheritance from his brother. Based on intelligence received from Eddard Stark before his arrest and execution, Stannis claims that Joffrey and his younger siblings are not Robert's children at all, but instead the product of incest between Queen Cersei Lannister and her twin brother, Jaime. This makes Joffrey a usurper and pretender, and leaves Stannis as the rightful king. How widely this claim will be accepted is unclear. Stannis commands a powerful fleet based at Dragonstone, but only a few thousand men are sworn to his cause.
King Robb Stark does not claim the Iron Throne at all, but has instead been proclaimed the King in the North by popular acclaim. He commands the combined military strength of both the North and the Riverlands and is currently engaged in fighting the armies of House Lannister, having already inflicted one significant defeat upon them. Robb's ambition is to secure independence for the North and Riverlands from the Iron Throne and to, if possible, seek vengeance and reddress from Joffrey Baratheon, whom he holds responsible for the death of his father.
The following factions have either not declared an interest in the conflict, or have already declared their neutrality:
House Arryn and the knights of the Vale have declared themselves neutral in the conflict. Lady Lysa Arryn, acting as regent for her young son, Robin Arryn, has stated that the Vale will hold its strength and not take part in the war, despite entreaties from her sister, Catelyn Stark, to join the Stark cause. Lysa is sceptical of Robb's ability to defeat the seasoned, more experienced Tywin Lannister in battle.
House Martell has pursued a course of isolationism since Robert's Rebellion, and has declared no interest or faction in the current conflict.
House Greyjoy of the Iron Islands has likewise pursued a more isolated course since their own failed rebellion against the Iron Throne a decade ago. With Theon Greyjoy still a hostage and ward of House Stark, it is unclear what their intentions are or will be in this conflict.
The Night's Watch plays no role in the internal political affairs of the Seven Kingdoms, and are sworn to defend the realm as a whole, no matter who is on the Iron Throne or competing for it. As a result, they are neutral in the current conflict.
Course of the conflict
The stage was set for the war when Catelyn Stark seized Tyrion Lannister at the Crossroads Inn and accused him of the attempted murder of her son, Bran. House Lannister summoned its armies and invaded the Riverlands with 60,000 men, striking at bannerhouses supporting Catelyn's father, Lord Hoster Tully, in reprisal. King Robert attempted to defuse the situation, but failed, and soon after died. Immediately upon hearing of his brother's death and having failed to win the support of the King's Hand, Eddard Stark, Renly Baratheon fled the city with Ser Loras Tyrell and rode hard for Highgarden. There, the Tyrell banners were called and began gathering, whilst Renly was crowned King of the Seven Kingdoms.
Prince Joffrey also immediately claimed the throne upon Robert's death. When Eddard tried to take Joffrey into custody as an imposter, Joffrey, forewarned by Lord Petyr 'Littlefinger' Baelish, had Eddard arrested instead. Unfortunately for Joffrey, he was unaware that Eddard had already sent a letter to Robert's brother Stannis, telling him that Joffrey was not legitimate and the crown belonged to Stannis by right. On Dragonstone, Stannis likewise claimed the throne and began gathering his supporters.
The first battles
Receiving news of his father's arrest, Robb Stark mobilised the armies of the North and assembled a host of 18,000 men (due to the urgency of the situation and the vastness of the North, he could not wait for more). He marched to the relief of the Riverlands. Even winning the support of House Frey and its troops could not bring Robb's forces up to parity with the Lannisters. However, the Lannisters had divided their forces into two formations of 30,000 men apiece, one under Lord Tywin's command designed to intercept the Stark army and the other under Ser Jaime tasked to capture Riverrun, the principle stronghold of House Tully. Robb sent 2,000 men to pretend to attack Tywin's army. As planned, this small force was defeated at the Battle of the Green Fork, but it delayed the Lannisters long enough for Robb's main army to slip past them and into the Whispering Wood near Riverrun. There, Robb staged a feint to draw Jaime Lannister and a portion of his army into the woodlands. During the Battle of the Whispering Wood, Robb inflicted a significant defeat upon the Lannisters, destroying Jaime's host and capturing the Kingslayer himself.
Receiving word of Jaime's defeat, Tywin realised that the Starks and Tullys were now free to unify against him, potentially bringing much greater numbers to bear against his smaller force. He retreated to Harrenhal, the formidable castle on the north banks of Gods Eye, intending to fortify it and use it as a base of operations to conduct operations in the Riverlands. By this time news of Renly and Stannis gathering armies had reached both the Stark and Lannister camps. After considering joining forces with one of the brothers, the Northmen and Riverland lords chose the path of independence, rejecting the authority of the Iron Throne all together and swearing fealty to Robb Stark as the King in the North.