iPhone 

see Apple support page for full details

iPhone X - Press the side button and the volume up button at the same time 

iPhone 8,7,6 - Press the side button and the Home button at the same time

Samsung Android

see Samsung support page for full details

Devices with a physical Home Button

Press the Home button and the Power button at exactly the same time. Once you hear the 'click' you'll be able to view and edit the screenshot. 

Press the Volume Down button and the Power button to take a screenshot. Hold both buttons at the same time and you should hear a click to indicate the screen capture. 

Palm swiping is another option but must be turned on your settings.

Google Android

see Google support page for full details

Press the Power button for a few seconds. Then tap Screenshot. If that doesn't work, press and hold the Power and Volume down buttons at the same time for a few seconds.