From Reddit I believe:
They are fighting an ideological battle for each other’s soul. They are both two sides of the same coin:
Two individuals who, because of a horrible day in their lives, became insane and decided to take on the world and make it in their image.
The joker was a ****ty comedian with a pregnant wife, a nice guy. On the day his wife died in a random accident he was bullied by mobsters into committing a crime, fell into a vat of chemicals and ended up alone, in pain, and scarred for life. The overall pain was such that he snapped…. realized that the world is cruel, unjust and random and decided he was going to destroy all fabric of the attempted, false, self-delusional order of the world and break everyone down to his level. He believes morals, ethics, are hypocritical nonsense. You can refer to the Dark Knight movie, in which he says “I’m just ahead of the curve.” He spends the entire movie putting everyone in front of him in situations where, to survive, they will have to break their moral code. Even the henchmen of the black guy… there are two. For no reason other than to break them, he says he will hire the one who will kill the other.
This is what the Joker does, he lives to prove to people that he is the avatar of who they really are : he just refuses to lie to himself.
Batman watched his parents be murdered, went insane also and developed several obsessions, he fights to bring justice to a world he feels is essentially good and plagued by the unnatural disease of crime and evil. He believes in justice above everything else, he does not kill.
So what happens when these two men face each other? The Joker’s ultimate victory is for the Batman, the strongest enemy of his world view, a person who refuses no matter what to break down to his level, to kill him. He wants the Batman to kill him. He can’t wait for Batman to do it. It will prove his point: anyone can be broken into evil, just like him, if their pain or their reasons are strong enough.
Meanwhile the Batman is facing someone who is the epitome of cruelty and senseless crime. He HAS to beat the Joker according to his rules, to prove to himself that his rules mean something, that they are absolute. And this is a decision he has to face every time he catches the Joker: do I kill him? How many lives will I save if I just kill him? He always escapes Arkham…. I will be doing a good thing by ridding this world of this supremely deranged psychopath. If only he could break his morals in this one case…. this one time… for the greater good….
The Joker knows this. And he laughs. And he hopes.
But he also has to deal with the temptation… without the Batman he would be virtually unstoppable. Even in the world of DC Comics where there is Super Man, other supervillains fear him. They steer clear of him. He is too unpredictable, chaotic, and cruel. If only he were to kill the Batman, there is nobody out there who understands him enough to be able to stop him. If only he could kill the Batman… everything would be so simple.
They are fighting a deeply personal, deeply ideological war. They each represent what the other one hates the most, and they each depend on the other to stay alive until the other bends to his will.
The last each one of them wants is to kill the other.
It is poetic.