How do you do a planche push-up?
A planche is a term that is usually used to describe a hand balance. The arms are fully extended, with the palms pressed into the floor, and the body (from toe to head, but excluding the arms) is held firm and roughly parallel to the floor.
A planche pushup, therefore, would be a pushup hand balance. The body would be held roughly parallel to the floor as it is raised up and down. (In other words, a planche pushup is like a normal pushup, but with the toes never touching the floor and with the hands more below the hips -- for balance -- than below the shoulders. Most people doing planche pushups tend to curl the feet inward, but I think that the best form with with legs fully extended.) Having said this, of course, I should add that I can't do these myself. I'm working on it, though. I have the balance and the overall stength but don't yet have the shoulder strength.
YouTube is a great resource for this sort of thing and often includes clips from people showing how they've trained to achieve specific tasks. I've included a couple of links below. It's amazing what people -- mostly amateur athletes -- can do!