For a long time, the only way to take a screenshot on an Android phone involved rooting the device (a process similar to jailbreaking an iPhone) and installing specialized software. Thankfully, that's no longer the case in Ice Cream Sandwich and Jelly Bean. Just press the power and volume-down buttons simultaneously. The screen will flash and a screenshot will be saved in your phone's photo gallery.
If you're on Froyo, Gingerbread, or Honeycomb, you might still be able to take a screenshot without special software -- sometimes manufacturers build screenshot shortcuts into handsets and tablets. If your device doesn't have this ability and you don't want to root it, try No Root Screenshot It -- you'll have to connect your phone to a computer to enable screenshots each time you restart the phone, but the app does exactly what its name promises.