With a Home button:
Hold down the Home button and the power button for a second and let go. Your iPhone will flash and make a click noise.
No longer have a Home button:
Press and hold at the same time the volume up button and the power button for a second, Your iPhone will flash and make a click noise.
To find it, head over to Settings, then General > Accessibility. Scroll down to AssistiveTouch. Turn it on and a small circle will appear on your home screen containing shortcuts to various actions, one of which will be taking screenshot.