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. 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, but you really don't want to root your device, 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.