- "I don't want to kill you. What would I do without you? Go back to ripping off mob dealers? No, no, no. No. You... complete... me!" -- The Joker, from Christopher Nolan's The Dark Knight
Simply put, an Arch-Enemy is simply a hero's singular, primary, and most important enemy. Among a Hero's Rogues Gallery, there will be one villain who stands head and shoulders above the rest. That villain will no doubt be the Hero's Arch-Enemy.
Try it yourself. Clear your mind, and then think, "Batman". What villain came to your mind first? Buddy, there is a reason the picture on this page features both Batman and The Joker.
An Arch Enemy can be a criminal mastermind or the criminal mastermind's evil lieutenant. He can be a rival, a worthy opponent, an evil counterpart, or even a mook who was promoted to name villain. The essential element that makes him different from the other villains is that, with him, it's personal. The villain fights the hero not simply because the hero has stopped his crimes and put him in prison, but rather because he hates the hero and wants to destroy him. The hero is seen as a personal affront to the villain (or vice-versa), so rather than merely capturing and killing the hero, the Arch-Enemy will take creepy photos of the hero's kids while they're at school, abducts the character's wife, seduce the hero's sister, kicks the hero's dog, and above all, leaves calling cards and clues to ensure that eventually he'll get caught.
In extreme cases, he'll stuff the hero's girlfriend into a refrigerator.
It is worth noting tha tbeing an Arch-Enemy has nothing to do with being the most powerful or most dangerous threat the hero has ever faced. Take Lex Luthor, for example. He is, without any doubt, Superman's Arch-Enemy. Of Superman's villains, Braniac is far smarter than Luthor. Darkseid, being a truly Galactic-level threat, is vastly more powerful and more dangerous. But for Lex (and, in truth, for Superman himself) it's personal.
An Arch-Enemy's relationship to the hero, and his motives for battling him, are usually in contrast to the way the rest of the Rogue's Gallery treats the hero. For example, the enemies of Reed Richards tend to have a great respect, perhaps even a reverence, for the hero's intellect; Doctor Doom, however, thinks Richards is an idiot. While the others often pick fights with Richards just because of this respect, Doom is usually out to prove to the world that his is the superior intellect, and that Richards is second rate at best, an utter fraud at worst.
Arch-Enemies are almost always the Foil of the hero. It is possible for a hero to possess more than one Arch-Enemy if more than one villain from his Rogue's Gallery manages to stand out. For example, with Spider-Man, the Green Goblin, Doctor Octopus, and Venom have all established themselves as iconic foes of Spider-Man, and each have been considered the wallcrawler's Arch-Enemy at different points in time.
When a one-off villain who is only intended to make a single appearance and then vanish becomes an Arch-Enemy by accident, that's The Moriarty Effect.