Until the Galaxy S8 ($749.11 at Amazon.com), taking a screenshot on one of Samsung's smartphones was done by pressing the power button and home button at the same time. It was easy to remember, especially for those switching from the iPhone, since it's the same button combo on iOS.
As you know, Samsung now uses a digital home button on the bottom of the display. That decision forced a lot of changes, such as moving the fingerprint sensor and using a new button combo to take screenshots.
To take a screenshot on the Galaxy S8, press the power and volume-down buttons at the same time. Pretty simple, right?
In the settings app under Advanced Features, there's a Smart Capture section that shows the ability to swipe your hand across the display as another method for taking a screenshot, but I've had no luck actually getting it to work.