Nowadays, an antihero is usually thought of as a badass, bitter, misanthropic, violent, sociopathic angry person (see Nineties Anti-Hero). However, this is actually a recent invention. For much of history, the term antihero referred to a character type that is in many ways the opposite of this.
In classical and earlier mythology, the hero tended to be a dashing, confident, stoic, intelligent, highly capable fighter and commander with few, if any, flawsand even fewer real weaknesses. The classical antihero is the inversion of this. Where the hero is confident, the antihero is plagued by self-doubt. Where the hero is a respected fighter, the antihero is mediocre at best. Where the hero is brave and courageous, the antihero is frightened and cowardly. Where the hero gets all the ladies, the antihero can't even get the time of day.
In short, while the traditional hero is a paragon of awesomeness, the classical antihero suffers from flaws and hindrances. The classical antihero's story tends to be as much about overcoming his own weaknesses as about conquering the enemy.
As time has gone on, this portrayal has become increasingly popular, as readers enjoy the increased depth of story that comes from a flawed and conflicted character. Hence, the classical antihero has to some extent replaced the traditional hero in the minds of readers as the idea of what a hero should be. It is nowadays rare to find a hero who does not have at least a little of the classical antihero in him.
See also Punch Clock Hero. Compare Super Loser and Tragic Hero. Contrast with The Ace and Nineties Anti-Hero.
Nozomu Itoshiki of Sayonara, Zetsubou-Sensei. AKA Mr. Despair, he is constantly attempting suicide and angsting about the most ridiculous of things. Interestingly, he isn't an example of This Loser Is You, as he's very good looking, intelligent, and comes from a very wealthy (if bizarre) family. In fact, the irony of his character is that he acts the way he does despite having these advantages.
Tatsuhiro Satou of Welcome to the NHK is a highly unstable NEET who places all of the blame for his highly unstable life on a conspiracy organization known to him as the NHK. And yet he is ultimately a good-hearted person who wants to be a productive member of society, most of his angst stems from feeling he is unable to lead a productive life.
Renton Thurston in Eureka Seven, who eventually graduated into a proper hero.
Early on, Vincent Law of Ergo Proxy is very poor material for a traditional protagonist; he's shy, awkward, holds little social standing, and works doing a very dangerous job. He considerably bulks up his credentials as the series progresses.
Akitsu Masanosuke from House of Five Leaves is a classical anti-hero, being an overly humble samurai with no self-esteem.
Natsume from Natsume Yuujinchou is a Socially-Awkward Hero with no self-confidence about people and a tendency to alienate what friends he does make by constantly lying to them to avoid causing a fuss.
Madoka spends most of the story struggling to cope with the horrifying things that happen to her friends, while being too scared to actually do much of anything (to the point of being The Scrappy to a lot of people). But she slowly overcomes her fears, and eventually summons the courage to become a magical girl in order to fix most of the tragedy.
Early Spider-Man, explicitly designed to be the first superhero with personal and internal conflicts besides super-villains and criminals. Spidey's runaway success was a major part of why such depictions came to be the typical depiction of a hero.
Evelyn Waugh's first novel, Decline and Fall, has Butt Monkey protagonist Paul Pennyfeather who is one of these in the way he is rather a pushover taken advantage of by the other characters.
Frodo Baggins in The Lord of the Rings, who ultimately fails in his mission to destroy the One Ring and is increasingly haunted by the physical and emotional scars of his journey throughout the story and for the rest of his life.
Gilbert Norrell of Jonathan Strange & Mr Norrell, while a skilled magician, is a humorless and petty character who is far from evil enough to be an Evil Sorcerer, but also far from sympathetic (or interesting) enough to be a traditional hero.
Lily Bart from Edith Wharton's House of Mirth. Let's see: fails at anything and everything she tries her hands at? Check. Only ever succeeds at alienating the few people who genuinely do care about her? Check. Is a whiny, insufferable Jerkass with an entitlement complex bigger than Brazil? Check. Dies at the end? Check.
Lola from Kit Whitfield's Benighted is pathetic, self-loathing and self destructive, turning away from or turning on anyone who might help her.
Mick "Brew" Axbrewder from Stephen R. Donaldson's Man Who series, a self-pitying alcoholic who makes Thomas Covenant look like Binky the Clown.
Linden Avery in the second Chronicles of Thomas Covenant trilogy. Becomes a more standard heroine in the third trilogy. Stephen Donaldson is very fond of taking classical antiheroes and transforming them.
Hank Thompson, the protagonist of Caught Stealing by Charlie Huston is this at first, a promising high school baseball player who wrecks his leg then , after a car accident where a friend of his is killed slowly spirals down into an alcoholic slacker, Then after inadvertently getting involved with conflicting criminal elements he levels up.
Live Action TV
Dave Lister, Cat and Arnold J. Rimmer from Red Dwarf start out like this, although Rimmer is both a neurotic loser and a smeghead. Lister once good-naturedly described himself as a "bum", while Rimmer would call him a lazy slob. Cat was vain, self-centered to the point of callousness, and not very smart... not surprising given that his species had evolved from a single, pregnant female housecat 3 million years ago (imagine the inbreeding), and even other cats considered him a moron. All three became more competent in the course of the series, but they never quite lost their essential quirks, their good qualities (such as Lister's selflessness and sense of fairness) merely became more pronounced. Or, in the case of Arnold Rimmer, who had no redeeming qualities, Rimmer had a run-in with his Alternate Universe counterpart "Ace" Rimmer.
Shinji Kido from Kamen Rider Ryuki is a good-natured buffoon who, for the majority of the series, is the only Rider attempting to stop the other Riders from killing one another. He never succeeds and for most of the series is plagued by his inability to save the Riders from destroying each other.
In the context of science-fiction TV history, Doctor Who was originally one of these. Pre-Who, space travel on TV featured handsome, youthful spacemen aligned with heroic, paramilitary forces. But the Doctor, at his core, is Jack Kerouac in space and time—a dropout from his own people who now just travels around like the '60s never ended. Also, in the William Hartnell days, Ian Chesterton was the male lead, and the Doc was a selfish anti-hero.
Michael Dugdale in Utopia is a rather hapless and borderline suicidal civil servant working for the UK's Department of Health and is blackmailed through various means by The Conspiracy into working for them to bring about a Sterility Plague. By the end of the series, he's broken into a potentially fatal quarantine zone to retrieve biological samples, stormed a Secret Government Warehouse with a shotgun and torched it, saved his wife and his marriage, brokered a deal with the conspirators to leave him alone and given a home to a little girl whose family was murdered.
Willy Loman from Death of a Salesman. A little, pathetic man, broken by his chase after a dream that isn't true.
Woyzeck from the eponymous play is considered the first true Antihero, as opposed to the classic tragic hero.
Everyone but Ricky Roma in Glengarry Glen Ross qualifies, but with particular attention paid to Shelly Levene.
Hamlet was conflicted and emotional before it was cool.
Travis Touchdown, of No More Heroes, a porn-obsessed Otaku without anything resembling a social life. He's also a Nominal Hero, however, eagerly slaughtering opponents and rarely showing any remorse for his killings.
Captain Martin Walker for most of the beginning of Spec Ops: The Line comes off as one of these his actions only cause disaster for both him and the people of Dubai. As the game goes on though it becomes more and more clear that he is actually a delusional Villain Protagonist desperately trying to be the hero of a situation far out of his control.
The post-scratch Kids in Homestuck eventually realize that, due to the symbolism of being in a void session, they are destined to simply sit around, get distracted by romantic subplots, and wait for the plot to continue without them.
Cherry from Cherry's Cure has physical limitations, is slow mentally, and knows all of these things. But is still the hero of the story. She's convinced she can't do what's asked of her, but needs to.
Aquerna, of the Whateley Universe. She is one of the Whateley Academy Underdogs, with laughable powers that make her a campus joke. She has self-esteem problems, and is no longer welcome in her own home since she turned into a mutant. Her combat final story and her Christmas story are all about her personal life and her personal problems, even if some action intrudes into the plot.
Every main character in Red vs. Blue qualifies on a comedic level, but a dramatic example exists in Leonard Church, who is a hilariously bad shot, can't seem to accomplish anything, and, in particular, constantly fails in what seems to be the only driving force in his life: being with his ex-girlfriend, Tex.
Taylor Hebert, of Worm, a bullied teenaged girl with cripplingly low self-esteem, who finds her escape in going out in costume. Her power is relatively weak (the ability to control insects), and her main victories come from working with other parahumans instead of defeating her enemies alone. Worm is as much about her growth as an individual as it is about The End of the World as We Know It.
Jaune in RWBY, who is the only main character not good at fighting and faked his transcripts to get into Beacon Academy. Most of his character arc involves resolving his shame over that. Incidentally, much of his interactions with Pyrrha exemplify how the two hero archetypes play off each other.
Cody and Sierra Total Drama World Tour. First season, Cody was a standard hero, but developed less heroic traits in the third season.
Fenton Crackshell from DuckTales was extremely good at counting, which is how he became Scrooge's accountant. But he often screwed up everything else, and he also was a dork, who had grown up in a trailer park. And when he gets a girlfriend, he becomes painfully hen-pecked. And yet, he was the super hero Gizmoduck, and he also saved the day four times without his Gizmoduck suit!