To capture a screenshot on an iPhone or iPad, press the Home button on the front and the Sleep/Lock button on the top of the device simultaneously. The screen flashes white when done correctly.
Your screenshot will save to the Photos app under Saved Photos or Camera Roll depending on your device.
For Android 4.0 and above, press and hold the Power and Volume down buttons simultaneously. You should hear a shutter noise and see the screen flash. A notification will appear at the top of the screen, from which you can access your screenshot.
You can also hold down the Power button until the option to take a screenshot presents itself at the bottom of the screen.