Nowadays with The Dark Knight having established itself in the eyes of many as being the best Batman film, Heath Ledger is in turn considered to be the best Joker. I’ve always thought that was more up for debate than many though. To me, Jack Nicholson was just as good, but the issue was that they played entirely different characters. Part of the issue with adapting a classic character like the Joker who has been around for 75 years now, is that there have just been so many versions of the clown prince of crime.
Heath Ledger’s Joker was crazy to the point that you could argue that he wasn’t even necessarily evil. As he said to Harvey Dent, he’s just like a dog chasing cars, simply doing things without terribly dwelling on them. He’s probably about 80% crazy, and 20% evil. He’s simply a force of chaos, and views the universe as being equally chaotic and thusly tries to impose that worldview on everyone else. This Joker is simply a homicidal sociopath with self-destructive tendencies, deep down wanting to die, as could be seen when he was urging Batman to take him out during their game of chicken in the street. I think that Ledger’s Joker, probably stemming from trauma in his character’s past, is deeply self-loathing and wants to die. This would fit with the Joker as seen in Alan Moore’s classic ‘The Killing Joke’ where he seeks to prove that all it takes is one bad day to break a man, with the implication being that he himself was broken. Though he tried to break Commissioner Gordon in the comic, Harvey Dent filled that role in the movie.
Jack Nicholson’s Joker was quite different. His character is probably 80% evil and 20% crazy. Seen as a well-educated career criminal with specialties in chemistry, he became the joker not through face paint and self-mutilation, but through falling into a vat of chemicals that disfigured him and made him crazy. This Joker is a vain narcissist with a fondness for art and power, as well as an overwhelming urge to show the denizens of Gotham that he is the best freak in town. This version of the character is more reflective of the classic Joker from the 40’s all the way until the 80’s when the movie was made.
The Joker was always smart and crazy, and he certainly killed people, but he also typically had clear goals just like Nicholson’s did, whereas Ledger’s didn’t. He just wanted the world to burn.