A smirk is specific kind of smile, one that suggests self-satisfaction, smugness, or even pleasure at someone else's unhappiness or misfortune.
Smirk can function as either a noun or a verb: "Wipe that smirk off your face. Don't smirk at me, buddy: you're gonna get yours next!" A smirk implies you think you're better than the person you're smirking at. Ever heard of the term "service with a smile"? Yeah, well, there's a reason it's not "service with a smirk."