Batman's Funhouse Mirrors: The Joker
To me, the thing about the Joker is that he's a reflection of Batman that helps illustrate what Batman is by being the same in some ways, and utterly different in others. They've both built themselves into symbols of fear, the kind of name that ordinary folks in-universe say with a hush, as if saying their name risks calling their attention -and thus wrath- down on you. Or you scream it in terror because you've just spotted them. Fear, is the point: Batman scares people. The Joker scares people.
The difference is in how they go about it, what they're trying to achieve, why they do it.
Batman deliberately chose to make himself into an icon of terror because a terrible thing happened to him that he doesn't want to ever happen to anyone else if he can help it. He wants to scare criminals straight: if you're legitimately terrified that The Goddamn Bat is going to drop in on you the instant you try to hold up a tiny grocery store, then maybe you'll go looking for some way to make your money that doesn't risk The Bat's attention. Furthermore, Batman doesn't really go out of his way to induce terror, per se. He selected his imagery to frighten, and he's no stranger to using fear to get what he wants out of people, but he mostly doesn't prioritize around the capacity of an action to produce fear. The primary goal is to stop criminals from being criminals, and fear is simply one of his tools, if perhaps the most prominent one.
The Joker revels in fear, and terrorizes people for its own sake. He's unpredictable, chaotic, disturbing. In some interpretations he opposes Batman philosophically by virtue of being convinced that any happiness is a lie and it's his job to tear down the façade -where Batman is trying to make the world a better place, the Joker is trying to convince people the world is worse than it seems. Other versions of the Joker simply enjoy the results they produce, that people screaming and running is its own reward, and any apparent end is an excuse, a justification. These Jokers don't rob a bank because they want the money -they rob the bank to provide a seeming structure to their fearmongering, a narrative for people to take some trust in... and for the Joker to then use against them. After all, if the Joker is robbing your bank, then you can cooperate with him and he won't shoot you, right? Ah, no, false hope: the Joker is here to shoot you (Well, to shoot someone, anyway), and robbing the bank is as good a cover story as any. It's a thing criminals do, right? The instant you think you're safe he'll kill you, surprise!... except when he doesn't, because if he did it every time, he'd become predictable.
Incidentally, this ties in quite nicely to his clown imagery, and not because there are people who find clowns disturbing. Fear and humor hinge on the same mechanism: uncertainty, unpredictability. A joke makes you laugh by surprising you. Fear is driven by the unknown, the lurking danger you think is out there but don't know when it will strike, where it will strike, why it will strike. Fear and laughter go together, so it's only natural that the Joker is a fear-mongering clown. The format of a joke is similar to the format of a nightmare -there's setup that you're not sure where it's going, and then there's a punchline out of nowhere. You laugh, caught off guard by joke, or you are struck down the very moment you think you're safe.
The result helps draw a line that helps defines Batman -it's tempting to think he frightens people for no better reason than because he enjoys it, but by contrasting him against the Joker -who most certainly does revel in fear for its own sake- the audience gets clarity that Batman isn't that way.
Next time, I'll talk about Robin.