For me, a compelling villain needs to be menacing and well-matched to the protagonist. Having a backstory is great, but only if it actually fits with the character and gives some heft to their villainy. Darth Vader, for instance, is a great villain (very menacing and basically Luke's shadow self once the whole father thing comes out) but the backstory Lucas finally gave us in the prequels was ... a bit lame (okay, a lot lame) though much of that was in the telling. With a better script and better actor playing Anakin, it might have worked.
Personally, my favorite villain (not sure if he's the greatest of all time, but he's up there for me) would be The Master from Doctor Who. I'm not up to speed on the new series incarnations (Derek Jacobi, John Simm and Michelle Gomez) but the original Master, Roger Delgado, was a classic (and Anthony Ainley, who took the part during the Fourth Doctor's tenure, was a pretty good successor). Delgado cut a pretty sinister figure to start with and the character was quite literally the Doctor's opposite: another renegade Time Lord but one who fell out with the Time Lords because of his lust for power rather than a desire to use his skills and knowledge to protect others. He could, depending on the writers involved, be a classic B movie mustache twirler but Delgado was a good enough actor that even in those stories, there was a bit more to the character than your usual cardboard cutout villain. And, while he never abandoned his lust for power, The Master was also a pragmatist who would cooperate with The Doctor when needed (but always to backstab when the opportunity arose, sometimes to his own detriment). I believe that the new series has expanded on the background, particularly having the two being friends back before they left Gallifrey (the Time Lords' homeworld) but, as I said, I'm not up to the season where The Master was reintroduced.
Oh, and there is one incarnation of The Master I tend to ignore: Eric Roberts in the 1996 TV movie was just awful. Which was too bad because Paul McGann was terrific as The Doctor and deserved a better villain.
I am very drawn to female villains with an overt sexual/sensual facet - obvious examples would be Harley Quinn and Mystique. Not only do they inspire creative bedroom activity (ha!); but I love how they create this lovely tension between being evil and enticing. I'm no hardcore feminist, but it's some kind of powerful to have a guy (unwillingly) sporting a hard-on whilst having his balls cut off...
As for the GVOAT? I'd have to give this to C Nolan's The Joker. Heath Ledger was perfection.
Batman has one of the best rogues' galleries of any superhero and The Joker stands at the top of the heap in that group. And when you get into films and TV, there are so many great portrayals to choose from, too: Romero, Nicholson, Ledger, Hamill (Mark Hamill voices the Joker for many of the DC animated series and movies). One of the complaints about the Marvel universe has been a lack of solid, interesting villains (aside from Loki). That should not be a weakness for the DC-verse if they do things right (alas, that has been an issue so far). Interestingly, they are looking at doing a standalone (ie. not part of the DC cinematic universe) Joker origin movie and have brought Martin Scorsese in as a producer, apparently planning to give it some of the gritty, street-smart feel of his classic crime movies.
I think "pure evil" is pretty lazy in terms of constructing a character. No one ever really considers themselves to be evil, it's always a label that is applied from outside the self by others.
As a writer, I truly believe you should love your villains as much as your heroes. The best villains are the ones who have motives that we can identify with. In psychoanalytic terms, a villain is a shadow character - a representation of the repressed parts of ourselves that we wish we could disown. They are the extreme unchecked impulses of the id and/or superego that are buried within us all projected and manifested in the 'other'. Interestingly, this means that stories of good and evil, heroes vs villains are in some ways psychologically fragmented and schizophrenic. The conflict is usually resolved when the forces of good dominate, control, suppress, or vanquish the villain. However, this is simply reenacting the repression which perpetuates the schizophrenia. A more healthy resolution involves acknowledgement, acceptance, and reintegration of the fragmented aspects of the self - the hero and villain unify.
Post-avant-retro-demelodicized-electro-yodel-core is my jam.
While the Batman villains, who are splendid in their psychological problems, most villains are the product of slowly breaking down their moral character. And almost any character can become a villain. Even if the 'good' protagonist is developed right, they should be villainous to a degree. Some of the best villains who morphed into their villainy over time are religious leaders, politicians, etc. Those who've lost touch with their original intent. A great villain is one who started good and becomes villainous.
