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From the first season, Jaime Lannister is part of the Kingsguard. He is the son of a very powerful family, but he chooses to stand outside the door of a king who he despises.
In season four we see that he a makes a deal with Tywin to spare his brother Tyrion,
But we are never really given information of why Jaime is so hell bent on being in the Kingsguard, what is the real reason for him being there?
Information from the books, as presented on A Wiki of Ice and Fire:
At the age of fifteen, Jaime participated in the campaign against the Kingswood Brotherhood, during which he saved Lord Sumner Crakehall from Big Belly Ben and crossed swords with the psychotic Smiling Knight. After the brotherhood had been defeated, Jaime was knighted on the battlefield by Ser Arthur Dayne, a member of the Kingsguard of King Aerys II Targaryen.
Now a knight, Jaime was to return to Casterly Rock. He decided to stop in King's Landing on route, to visit his sister, from whom he had been separated for years. There, Cersei informed him that Lord Tywin intended to betroth him to Lysa Tully, and had gone as far as inviting Lord Hoster to King's Landing, to discuss the dower. Cersei suggested that Jaime become a member of the Kingsguard instead, replacing the recently deceased Ser Harlan Grandison. Though this would require Jaime to give up his claim to Casterly Rock and his position as Lord Tywin's heir, it would allow him to be close to Cersei and free himself of the unwanted marriage to Lysa. After a night of passionate sex, he gave his consent to Cersei's plan. While the siblings expected Tywin to oppose it, Cersei claimed he could not openly object. She promised to do the rest, and a month later, Jaime received news at Casterly Rock that he had been chosen for the Kingsguard, and was to present himself to the king at Harrenhal during the grand Tourney, where he would say his vows and don his cloak. Jaime's and Cersei's plan did not entirely work as expected. Their father, furious about the King's decision, pleaded illness and resigned his position as Hand of the King. He returned to Casterly Rock, taking Cersei with him, which separated the siblings once more.
King Aerys II Targaryen went along with the plan for reasons of his own:
Jaime was raised to the Kingsguard by Lord Commander Ser Gerold Hightower in a ceremony during the tourney at Harrenhal in 281 AC, making him the youngest knight ever to be raised to the order. That night, however, King Aerys II Targaryen soured the honor by sending him back to King's Landing to guard Queen Rhaella and Prince Viserys, depriving Jaime of the chance to participate in Lord Walter Whent's tourney. It was that moment during which Jaime realized that the king had only chosen him for the Kingsguard to slight Lord Tywin, of whom Aerys was insanely jealous, and to rob him of his heir.
However, after his sister married Robert Baratheon their plan did succeed:
Jaime was part of the honor guard that brought Cersei to King's Landing for her wedding to Robert. Cersei and Robert were married in 284 AC, with Jaime present. He won the tourney held to celebrate the wedding. Jaime and Cersei had sex on the morning of Cersei's wedding, and continued their illicit relationship after Cersei's wedding. Cersei refused to join her new husband on his hunting trips, as to be able to spend time with Jaime. Jaime accompanied Robert and Cersei to Greenstone, when Robert went to visit his mother's family. Jaime dubbed the castle "Greenshit" when he first saw it, and convinced Cersei to do the same. They spent a fortnight with the Estermonts. During one of those nights, Jaime followed Robert on Cersei's request, to confirm her suspicion that Robert was sleeping with his cousin. Jaime and Cersei slept together that night, and Cersei likes to believe that their eldest son, Joffrey, was conceived that night.
An important note WRT the Kingsguard:
They are sworn for life
