It is possible to capture screenshots on a Windows Phone. You can do this by simultaneously pressing the power button and the Windows logo button on the phone. It may take a little getting used to to press them simultaneously. Unfortunately the screenshot sound is always played and doesn't obey to the sound settings like vibrate or no sound settings. Captured screenshots are stored in the photo album named "Screenshots" which is accessible via the Photos App.
With Windows Phone 8.1 they have changed the buttons that need to be pressed in order to capture a screenshot. They have now become the on/off button and the volume up button. So press these two at the same time to capture a screenshot. It now also no longer makes a sound when capturing a screenshot.