An antihero is a protagonist who lacks many of the standard qualities of a hero, sometimes being simply ordinary, sometimes subverting generic conventions, and sometimes being a completely despicable character.
One of the most frequently discussed antiheroes of modern fiction is the narrator of Notes From Underground, a novella by nineteenth-century Russian novelist Fyodor Dostoevsky. The nameless narrator opens the story with the claim:
I am a sick man.... I am a spiteful man. I am an unattractive man.
He describes over the course of the story how he became a hypochondriacal, peevish recluse.
Holden Caulfield, the protagonist of The Catcher in the Rye by J. D. Salinger, is often considered an antihero. He has just flunked out of school, and is troubled by a generalized sense of angst and frustration with what he considers the "phoniness" of the world around him. This realistic portrait of the psyche of a troubled teenager is decidedly antiheroic, and although many readers sympathize with Holden, he is does not fit many of the conventions of heroism.
Finally, Becky Sharp of Vanity Fair by William Makepeace Thackeray is an antiheroine, who is selfish and unethical, trying to claw her way out of poverty by means of her wits and her looks. Although we may admire her grit, she is the sort of character more commonly cast as an antagonist than a protagonist.