Widely regarded as one of the greatest villains ever created, The Joker is a rule all unto his own. His precise origin is unknown, as is his true identity. There have been numerous different tellings on his backstory; and all of them have a place within canon it seems, for The Joker himself has no real grip on his past, remembering it differently all the time. Indeed, he and others have claimed he reinvents himself each day.
He is Batman's greatest foe and responsible for causing some of the greatest harm to Batman's life.
 Completely insane, entirely sociopathic, brutally homicidal and violently unpredictable, he is the one villain even other villains are afraid of and reluctant to cross in any way.
Although he has no "super" powers (though an 'ordinary' human he is far from) he has committed such atrocities and is known for being so capricious that his reputation far precedes him, with the result many will flee from him upon sight.

The chemical bath that so altered his appearance - giving him chalk white skin, moss green hair and a rictus-grin - also did imbue him with other properties. He has a seeming immunity to most toxins and poisons and medicines as well as a heightened resistance to pain, and perhaps an enhanced healing factor (the number of times he has been subjected to normally fatal situations to merely follow with a full recovery suggests this possibility).

Joker seems to (or at least, pretends to) not consider anyone to be "real" but himself and Batman. He often claims the universe is his own creation and Batman is there to keep him from getting bored. He considers Batman his perfect foe and whilst he often pays lip service to the desire to destroy Batman, Joker prefers the Dark Knight alive.  Many speculate, and it has been hinted within the comics, that Joker actually knows Batman's true identity, but protects it in order to continue to have someone to play with.  Batman's ability to stop Joker's schemes indicate to Joker he is indeed the perfect foe and he protects Batman fiercely from other villains, considering Batman to be "his" and his alone.

Despite this, The Joker basically lacks motivation for his crimes, claiming they are all "a joke". He views life as a theatre of the absurd, the petty struggles and strivings of man to be utterly ridiculous and enjoys exploiting this with his sick sense of humour. Part of Joker's horror and ability to cause fear is that he is actually unafraid of death, if it helps him meet his ultimate objective. He was once willing to die in order to prompt Superman to take a life, and has often goaded other super-heroes to do the same; thinking it hilarious that those who've sworn never to kill could be driven to kill him and thereby come down to his level. He has a body count literally in the thousands and has no compunction about who he kills, choosing his victims indiscriminately; though it may be said he has a preference for those viewed as pure and virtuous by common society.

His crimes against Batman begun with the crippling of Barbara Gordon, the first Batgirl. Shooting her in the spine he then stripped her and took photos in the acclaimed Killing Joke one-shot. This was followed sometime after by the bludgeoning to death of the second Robin, Jason Todd, in the four-part Death in the Family series. Finally, he shot Sarah Essen-Gordon, the Commissioner's second wife, at point blank range after she attempted to stop him slaughtering thirty-six newborn children in the No Man's Land saga. He has been directly responsible for the deaths of numerous others involved in some way with Batman's life.

Joker is often underestimated by heroes and villains alike because he has no "super" powers. Whilst a fairly lean man, he possesses some fighting skill and possibly insanity-induced adrenalin rushes lends him strength. Coupled with his absolute unpredictability, these factors all combined means Joker gains the upper hand in most situations with a minimum of ease. In 'Devil's Advocate' an inmate who easily outweighed Joker by well over a hundred pounds ended up beaten to death with Joker's shoe for underestimating him. Likewise, the almost indestructible Marvel villain Carnage was bested by The Joker before ending up overcome when Joker was about to release a toxic gas poisoning everyone in the local vicinity, including himself, to prevent Carnage from killing Batman.

Joker has a genius level IQ and is a master of chemistry, developing his own trademark Smilex toxin - a poison that prompts people to laugh as they die, and causes the entire muscle system to go into rictus, resulting in a macabre death-grin. He also has great skills with genetics and nuclear engineering as well as being knowledgeable with computers. Additionally he is highly astute and perceptive and is a master manipulator and fond of psychological warfare.

There has been much speculation over Joker's sexuality over the years. It is generally agreed Joker is almost entirely asexual, having little or no interest in sex and no particular sexual identity.  Nonetheless, he is unafraid to camp it up, and indeed, very much enjoys this particularly where Batman is concerned, fond of making lewd innuendo. As he sees Batman as his flipside, he is in a way, enamoured of the Dark Knight because of the challenge he provides him with. He also knows the allusions and suggestiveness grates on Batman's nerves, which is something he loves to do.
With Harley, Joker can frequently be observed to engage in acts of affection as often as acts of violence, calling her pet names and referring to himself as 'Daddy' towards her. There have been allusions and hints as to their (admittedly infrequent) sex life. Many fans believe Harley actually awakened Joker's long slumbering libido, much to his shock and probable irritation. Joker is not the sort who would ordinarily tolerate someone like Harley for very long, if at all, but it is believed that her unexpected infatuation with him surprised The Joker with the result he kept her around as a curiosity and then, despite himself, began to have feelings for her - as much as he is capable of, anyway.  However, being The Joker, this makes little difference to his treatment of her and even his willingness to kill her, for whatever reason he deems appropriate. .

Joker is highly egotistical and narcissistic. It is said that if he truly loves anyone, it is himself only. narcissistic. It is said that if he truly loves anyone, it is himself only.

LINKS

Joker's page on wikipedia - lots of awesome info!
Joker's page on DCdatabase - again lots of great info, including some further speculation into his past and a more extensive vital stats. It also has profiles on The Joker's different incarnations during the Gold and Silver Age (Earth 1 & 2), accessible here.
Joker's page on DCAnimated wiki - great look at Joker in his animated form
List of comics Joker has appeared in (fairly extensive, but unsure if it's complete)
Joker over the years - a fantastic article on Joker's many incarnations and handlings over the years
Chelsea Grin - an awesome new site dedicated to Mistah J with a focus on Heath Ledger's interpretation of Joker
The Ha-Hacienda - a Joker fansite
Joker's Circus - a great fansite, though it hasn't been updated in a long time
The Killing Joke - one fan's criticisms
Scans_daily - a great post dedicated to Joker. Not dial-up friendly!
The Joker Problem - another great Scans Daily post exploring good Joker writing vs Bad