To have a bowel movement. Poop. Basically, do a "number two".
The Dick Cheney reference is derived from the fact that the President of the United States is number one and rides Air Force One. The Vice President is called number two and rides in Air Force Two. Plus, the Bush administration is shit... so there you have it.
Maria: "Aw man, what is that terrible smell? What did you do to my bathroom?"
Paul: "Sorry honey, I had to drop a Cheney. That bean burrito really did a number on me!"