The Kingsguard is a group of elite knights who are meant to protect the king of the Seven Kingdoms. Like the Night's Watch, the Kingsguard members are supposed to give up their families and devote themselves to their new job (which doesn't even have benefits). There are supposed to be seven Kingsguard men at a time, and they're all supposed to wear white (symbolizing their… purity? Or just to show off how clean they can get clothes?).
Except for Barristan Selmy, who is the old leader of the Kingsguard, and Jaime Lannister, who becomes the new leader (despite being known as the Kingslayer), it's not always easy to tell these knights apart. For instance, Boros Blount isn't very interesting as a character in this book, so it's not clear what makes him so different from the other boring Kingsguard knights: Arys Oakheart, Mandon Moore, and Preston Greenfield.
Actually, other than Barristan and Jaime, Meryn Trant might be the Kingsguard with the most personality. Sadly, that's because he will do anything he's commanded to, including beating up Sansa Stark. So, maybe it's better to be boring.
Considered the greatest living knight, Selmy has been in the Kingsguard through three kings' reigns. This means that he served the last king, Aerys Targaryen, and now serves the king who overthrew Aerys. (Sounds weird, right?) The important thing is that Selmy keeps his word, which is why Robert trusts him.
That's also why it's so surprising (and dumb) when Cersei and Joffrey dismiss him. Unlike some of the other knights in this book, Selmy seems to take the ideals of chivalry seriously; for instance, he votes not to assassinate the Targaryen children. That's pretty nice of him, we think. So what is he going to do now that the Lannisters threw him out? Will he just be happy enough vacationing on some island? Or is he going to cause some trouble for King Joffrey?