A very, very horrible thing. Don't ever ever do this even as a joke or anything. I node this only for the completeness of the database...
To curbstomp a person involves forcing them to lie face down on the road, and then having them put their mouth on the curb. Once in that position, if you bring your foot down hard on the back of their neck you will break their teeth, dislocate their jaw, and cause serious damage to their neck and spinal cord very probably killing them. That is a curbstomp.
Most people know of curbstomping from the movie American History X. In it Edward Norton's character, a neo-nazi white supremacist, forces a black man to lay down on the road and assume the necessary position. I don't actually know if we see Ed's foot come down as I had to turn away at that point. In any case, Ed is arrested for manslaughter and sent to jail, which is what will happen to you if you ever do this.