How to take, and view, a motion photo
It's just like taking a normal photo, but if the camera's "Motion" setting is on, your phone will capture 3 seconds of video as well.
When you look at a motion photo using Google Photos, you'll see a play/pause button at the top. If it's on pause, it will look like a normal still photo, if you click play, you will see the motion looping over and over.
What to do with a motion photo
There are lots of things you can do with motion photos, but I think the most important is finding a better frame for the still shot. Using Google Photos (Android) you can just swipe up on the photo, then swipe up further and you should see a filmstrip of "Shots in this photo." Tap there, find the shot you like, and tap "Save copy" at top right.