I would define a Villain, not necessarily as a character who does evil (though it does clear the issue somewhat), but as a character who uses any ideology and takes it to the most extreme point, and then commits acts like genocide, mass murder etc. under the banner of his ideology and tries to pass it off as a 'necessary' deed.
A villain is not the protagonist's opponent, nor is he simply the character seen opposing the audiences views. A villain is a character who does evil, is aware he does evil, and attempts to justify it by things such as 'Pay evil unto evil', and 'Utopia justifies the Means.'
Alternatively, discuss what is a Hero?
In the end I'll be with you.