Can you play a game on monitor and have your desktop up, or another window open on a second monitor?

I had dual displays working just fine. I tried everything I could think of when my friend left his monitor over after a LAN party, but I couldn’t get a game on one, and something else on the other. They were different resolutions… could that be part of the problem? Or is it not possible to do? If it is, how? And how would you switch between the game and desktop?