Hi, I'm the Slayer, ask me how!

In every generation there is a Chosen One.
She alone will stand against the vampires,
the demons and the forces of darkness.
She is the Slayer.

SLAYER: noun, Old English. Person who kills violently or with great bloodshed.

The Show revolves around the persona of the Slayer, a young heroine with the sacred mission of saving the world. The Slayer, a teenage girl, is endowed with great strength and ability. She fights the forces of darkness, protecting humans and preventing apocalypses. In theory she must fight alone and keep her mission secret. She is initiated and trained by a Watcher, who is part of a Council that has a high knowledge of the occult.
The Slayer was first brought to screens in 1992 by Joss Whedon and the movie 'Buffy the vampire Slayer'. It wasn't a big success, but the creator didn't give up. He re-worked the concept and turned it into a TV series in 1996. Buffy was a change in the usual horror flicks. This time no blonde chick getting killed by some evil monster in a dark alley; better, the girl actually fights back and kicks the monster's butt!

Buffy arrives in Sunnydale, trying to escape a troubled past. But Giles, the school librarian, and also Buffy's watcher, reminds her that she has a sacred mission, she's the Slayer. She soon blows her cover and finds out that several people in Sunnydale know who she really is. She befriends Willow & Xander, who help her fight evil, much to the Council's despair. She is even joined by a 'good' vampire, Angel.

A Slayer hunts vampires. Buffy is the Slayer, don't tell anyone. That's all the vampire information you need.
    Giles, The Harvest.

The training of a Slayer is fixed by the Council. She gets a handbook and must pass various tests. But as soon as he meets Buffy, Giles knew she wasn't like other Slayers, and that the set training process wouldn't work with her. He becomes much closer to her than he should, developing a father-daughter like relationship with her, compromising his loyalty to the Council and eventually getting himself fired.
Buffy soon starts rebelling too. Disgusted by their lack of consideration and the fact that they refused to help her save Angel, she quits them, and does things her way.

A Slayer has many abilities that allow her to fight evil. She is much stronger than your regular human. She is fast and agile. Her senses are highly developed and she has the capacity to recognize vampires even when they are under human form. She also heals very fast and barely bruises. But a Slayer's dangerous lifestyle means they rarely live over the age of 25, if they even get there.

Technically there should only be one SLayer at a time. When one dies, another one is called. Some seemed to be trained in advance, when their powers are detected early enough, like Kendra and the SIT's (Slayers In Training), but others are found out later, like Buffy. But in both cases, their full powers only appear when the previous Slayer dies.
In season one, Buffy drowns. Xander saves her using CPR, but she was actually clinically dead during a minute, and a new Slayer, Kendra, is called, leading to a unique situation: two active Slayers at the same time, neither of them knowing there was another and almost killing each other.
Kendra is a far more obedient Slayer, working by the rules, but nonetheless bonds with Buffy. She is killed by Drusilla, one year after her calling. Faith, a rather wild and unstable young woman, becomes the Slayer. Her own demons and the fact that everyone considers her as the 'second slayer', make her go bad and turn on Buffy. She later ends up in prison fur murder.

There are many references to former Slayers.

Buffy's first death disrupted the Slayer lineage, but her second death and more so her rebirth did too, putting herself, the SIT's and the entire world in danger.

So the is the Slayer. You're prettier than the last one.
    - The Master, Nightmares.

I knew a Slayer in the '30's. Korean chick. Very hot. We're talking muscle tone. Man, we had some times.
    - Sid, The Puppet Show.

There was this one Slayer, during the Boxer Rebellion...
    - Spike, School Hard.

Considering the average life span of a Slayer, Buffy has surprisingly reached the age of 22, although, technically she died twice. This also makes her question her mission and the true nature of the Slayer. She is disturbed by hearing that her powers come from the darkness and that that's where she belongs (First Slayer, Dracula, Spike). She is told that her true nature is no different that that of a vampire, that she is in fact a killer.
When she is severely injured by a regular vampire, Buffy realizes that she is vulnerable and tries to find out how short her life will be. A few months later she kills herself when closing a dimensional portal. Brought back by her friends, she remains obsessed by death and seems pretty detached from the world.

Being super heroes, another question arises. How far can a Slayer go? When does she abuse her powers? Can she take a human life?
Buffy and Faith have both faced this question and have opposite opinions.
Buffy killed her mothers boyfriend, Ted, by using her strength to fight him and push him down the stairs. She is deeply traumatized, but admits that when she pushed him, she fully knew what she was doing. The gang soon finds out that Ted was actually a robot and did indeed intent on hurting Buffy and her Mum.
Years later, Buffy was tricked into thinking she'd killed a girl during a fight. Upset, she prepares to turn herself in to the police, but realizes she was setup by Warren.
When Faith pulls Buffy into her whirlwind lifestyle, Faith accidently stakes a human under Buffy's eyes, mistaking him for a vampire. At first, she tries to keep it secret, claiming that it doesn't matter given all the lives she's saved. But when evidence starts showing up, Faith tries to pin the murder on Buffy. No one is fooled and Angel tries to reason her, but Wesley steps in and screws everything up, firmly pushing her over the deep end. She turns into a wild killed and starts to self destruct. Buffy condemns her actions and claims that their powers do not allow them to take a human life, but when she is faced with the decision of killing Faith, Buffy shows no doubt and no scruple. Needing her blood to save Angel, Buffy remorselessly stabs her and pushes her off a building.