An anti-hero is a protagonist (the "hero" of the story) who has qualities we would normally associate with bad guys or people who are the opposite of a hero. Because of these abhorrent traits, you would think that we, as moral human beings, should hate the character, but the author writes the story such that we empathize with them instead (even though a part of us probably does hate them a little.)

There are some great TV anti-heroes. For example: Walter White, main character in "Breaking Bad," is a drug (crystal meth) manufacturer and murderer who dissolves the bodies of his victims in hydrofluoric acid and is really mean to his young, 20-something partner, Jesse. We empathize with him because (1) he's dying of cancer, (2) he's just trying to make money for his family, and (3) the people he kills are bad guys, too. But by the end of the series, we have to admit we're a little disgusted with him (and with ourselves for rooting for him all those seasons.)

Other examples: Dexter, Tony Soprano, and even House was an anti-hero. He didn't murder anyone, but he was always just a big jerk.

Cheers,
Michelle