Depress as simultaneously as possible, and keep holding down for about a second, both the home button and the power button.
The order in which the buttons are depressed does not matter, but simultaneity matters: from the moment that one of the buttons has been depressed, there is an extremely small window of time, perhaps as small as one tenth of a second, within which the other button must also be depressed, or else no scweesho fo joo!
You will know that you have managed to take a screenshot when your phone will emit an oh-so-vintage camera shutter sound, a bright white frame will momentarily appear along the borders of the screen, and a "picture" icon will take seat in the notification area.
The oh-how-much-I-love-writing version of the post: