A sexual act likely performed with a hooker, stripper or some bar slut you just brought home with you.
Mix a martini up and bend your girl over in a doggy style position. Proceed to pour the martini down her ass crack whilst lapping it up in the vaginal/taint area.
When a girl gets a bottle of grey goose and sticks the end up her vagina then does a hand stand to get it all in there....thats what you call a dirty martini
When they ask for a dirty martini, you put the bottleup your butt and then pour some in and start twerking to stir it up, and then shotgun blast poop and alcohol all over their face.