Pressure flips were really popular in the early 90's, but people realized that when you did them too often, your board got beat up. Some people still do them, and they look pretty cool when done correctly though.
A pressure flip actually looks just about exactly like a varial heelflip. That is, your board rotates about the long axis away from you while doing a backside shuvit. The difference between the tricks is how you accomplish this.
With a varial heelflip, your back foot goes through the motion of a shuvit, and your front foot goes through the heelflip motions. Simple enough, right? For a pressure flip, all of the motion in the board comes from the back foot.
When you begin to pop your deck up, you start to do a shuvit, thrusting the rear of the deck around to your backside. The only difference is, you put a little pressure (hence the name) on your toe as you thrust. What you are effectively doing is giving your skateboard a little backspin, just like you would if you were shooting a basketball. You have to give it just the right amount, so that it only spins once though. Once the board has flipped over and spun around, you land with your feet on the bolts and ride away like a champ.
Note: This node was originally written in response to one that described a pressure flip as another trick entirely. This forgotten trick has been referred to by some misinformed skaters as a pressure flip, but is more commonly known as an old-school kickflip. It is a kickflip without the ollie motion; just flipping the board over quickly in a push-pull motion with the feet on the sides of the deck. The old kickflip was used in freestyle competitions for years, before Rodney Mullen invented the modern day version.