The Watcher's Council
The Watchers Council is an obscure secret society based in england. the Watchers are the ones chosen to train and guide the Slayer in her task: fighting evil.
They study the occult and mystical matters, but generally have another, more common job. Giles was once the curator of a british museum, or th British Museum, Willow wasn't sure, and then a high school librarian.
The Council hasn't really modernised over the years, and the watchers are still very old-fashioned. They have established strict rules on the training and the behavior of the Slayer. She must devote herself totally to her mission, train regularly and study the occult. But upon meeting Buffy, Giles realised that she was a different, rather undisciplined slayer and decided to put the rules aside, because he knew she would never refer to them.
If Buffy is a rather peculiar Slayer, Giles is also a peculiar watcher. At first he try to control Buffy, but he soon gave up. He is always around to help and developed a very close relationship with his protegée over the years.
Giles opposes himself to the Council's methods. In 'Helpless' her strongly contests the Cruciamentum, an ancestral test that the Slayer must pass when, or should I say if, she reaches the age of 18. During this test, the Slayer looses her powers and must fight a vampire alone in a closed space. Giles feared for Buffy's life and told her what was going to happen. When the Council found out, they fired him and sent in a new watcher, Wesley Wyndam-Price, rather incompetent, who needs Giles' help for pretty much everything. Wesley was never really accepted by Buffy and Faith, or by the rest of the gang, and finally left Sunnydale when Buffy quit the Council.
The Council also has rogue watchers, like Gwendolyn Post. She shows up in Sunnydale in 'Revelations', posing as Faith's new watcher, and pretty much tries to kill everyone, building Faith up against Buffy. As the two fight, she makes the most of it to get her hands on some ancient power and tries to set it free. She most likely played a big part in Faith turning bad by making her feel inferior to Buffy.
The Council doesn't seem very efficient. The first team they sent to capture Faith was pretty useless and failed shamefully. The second was a bit better, but still no match for Buffy (in Faith's body). They send the Slayer out to fight practically invincible occult forces on a daily basis and when it comes to bringing a rogue Slayer back to England, they send a team of three guys!
They seem to be more like a group of old Englishmen who spend their weekends playing golf, talking about the occult, thinking they're so much cleverer than everyone else.
But with Faith in jail and Buffy telling them to get lost, the watchers are pretty useless. When they find information on Glory, they think they're back in the game and try to force Buffy back to them by threatening Giles. Buffy reacts, realising that she has the power, and accepts to 'collaborate' with the Council but under her circumstances only.
The Council seems to be a family matter? Giles mentions that his father and his grandmother were watchers and that he 'inherited' the mission. Like Buffy, he didn't accept his destiny at first and was a rebellious teen.
The Watchers must also study the occult: demons, mythology, magic, ancient languages (Giles speaks six or seven), a female watcher even mentioned she wrote a thesis on Spike (who was very flattered when he found out!).
The process that allows them to find new Slayers is unknown, but the watchers often refer to prophecies, many of which can be found in the Codex, that also foretold Buffy's death in 'Prophecy Girl'.
During season 7, the Council's headquarters are destroyed by an explosion. Most of their works are destroyed and many watchers must have died. It's reasonable to believe it will take them a while to get themselves back together after that.