This is probably going to be a popular answer, but my favorite kind of villain is the one you can identify with. One you sympathize with. One you maybe even sort of agree with.
To me, one of the most dangerous and compelling villains is one who could be a hero, if only his/her methods or motivations were slightly different. In some cases, the line between hero and villain can be quite blurry. Exploring where a character tips from hero to villain can lead to fascinating discoveries about characters.
A recent example of this type of villain is Magnito in X-MEN: FIRST CLASS. Xavier and Magnito were friends and shared a lot of the same hopes for mutants. They got along; they'd been through a lot of plot together. But not only did they have different ideas of achieving their goals and what kind of losses were acceptable, they had different outcomes in mind as well, with one wanting peace and equality, and the other wanting to replace humanity with mutants.
In spite of their similarities, and the likability of both characters, one turns out to be a villain.
That said, the absolute evil variety can be interesting when they have a personality. (Think Angelus from BUFFY.) Or absolute evil with one redeeming quality. (The Mayor with Faith from BUFFY. Now that I think of it, that show had some fab villains.) Absolute evil is scary. It's looming. It's dark. There's no reasoning with it. You never know what it's going to do next. WATCH OUT IT'S GOING TO GET YOU!
But the villain you kind of like . . . Well, what does that say about you?
That's what I find interesting.