As is the case with many of life's answers is: it depends. Devices running Android 2.3 or earlier don't have a built-in way of taking screenshots. In Android 3.x, there are ways, but nothing straight-forward. In Android 4.x, however, things get a lot easier.
Take a Screenshot on Android 4.x
Press and hold the phone or tablet's wake/sleep button and the volume down button at the same time.
Within a few seconds, the OS take's a picture of the screen and places it in the photo gallery.
Open the photo gallery to see the picture.