For those of you who have never heard of him, Wheeler will probably be most remembered by the general public as the one who invented the term "black hole" for a dead star so dense that not even light could escape its gravitational pull. Oppenheimer and Snyder had suggested this possibility out of Einstein's general relativity, and it was at a conference in 1967 that Wheeler came up with the term.
The concept of a star so massive that not even light could escape had been discussed long before the equations of general relativity suggested the possibility, but no one had come up with a good term for the idea. Probably the most well-know phrase before "black hole" was "frozen star," which doesn't quite create the same image in the mind as "black hole" does.
Black holes have become a longtime staple of science fiction; I even used one for my first cover story, "Escape Horizon" (Analog, March 2000).
As someone who studied general relativity as a graduate student, I used Wheeler's classic co-authored textbook on the subject: Gravitation by Misner, Thorne, and Wheeler. It's one of the clearest explanations of general relativity for the physicist that I have ever seen. I also learned some of special relativity out of the classic Spacetime Physics book that Wheeler co-authored with Edwin Taylor; and although I did get to meet Taylor once (when I almost served as his Teaching Assistant), I never did get to meet Wheeler. I wish I had; I understand he was a great teacher. Wheeler was probably the most influential physicist of the 20th century who never won a Nobel Prize, and he deserved one a thousand times over.
If you want to learn more about him, here's a link to his New York Times obituary: John A. Wheeler, Physicist Who Coined the Term 'Black Hole' Is Dead at 96.