The Joker.
He's got everything:
Cool design that works perfectly for him and isn't "too busy"
(ie, they didn't decide to just give him an arsenal on his person or something)
Interesting method of killing
(They die poisoned, smiling, and in fear at the same time. Anytime they have an image of someone smiling wider than they possibly can whilst begging for help, that's just awesomely freaky)
You love to hate him
(I think everyone that likes this character laughs at his antics, but at the same time, registers that they probably shouldn't find it funny.)
He's the perfect foil for Batman
(Batman is a strict, dark, hardass that instills fear in others despite protecting people. The Joker is a fun, bright, playful guy who...despite hurting people, in a subtle way instills happiness.)
He's an IMPORTANT foil for Batman
(As Joker's stated numerous times, he'll never flat out kill Batman and Batman will never flat out kill him, cause its way too fun. Without the Joker, the Batman isn't as much of a protector, and without the Batman, the Joker isn't challenged much. They need each other and are at a bit of a sibling rivalry.)
He's sympathetic, though you have no reason to sympathize cause he's so evil
(If you look at him deeply, you can see that he's essentially wanting to make people laugh but since he has a twisted sense of humor, nobody seems to "get the joke". Also, he's like a little kid with ADHD. Any attention is good attention. The Joker wants respect and he gets it through being Batman's #1 foe.)
He has such a clear cut identity, but has no set backstory or even real name
(Everybody knows at least the basics about The Joker, but not even people that have read every single story he's mentioned in know what his real name is. How many villains have been there for decades and haven't been given this treatment? Some of them have even been revised to give them more than one. Look at Luthor lol. Mad scientist, now he's a businessman, now he's a businessman that once was friends with Clark, etc)
There's a lot of unique characters (and even more characters that try to copy from them) but I think The Joker takes the cake here. He's unique in the way that he's absolutely essential and perfectly sculpted. Nothing is forced at all. Its just a natural flow.