Taking a screenshot with the new Apple Watch is very easy: press the Digital Crown and the Side Button at the same time. That's all there is to it! Our own Adam Christianson found it when he was awarded what he called his "mystery 'You've achieved your Move Goal award," shown in the image below the diagram. Congrats on the mystery award, Adam, and thanks for the tip!
Press both of these buttons at once to take a screenshot
Apple Watch 'You've achieved your Move Goal' Award
Once you take your screenshot, the image is sent directly to the Photos app on your iPhone—it is not stored on the Apple Watch itself. Jim Tanous at TekRevue helped me test what happens when your Apple Watch isn't in range of your iPhone (or it is turned off), and we found that the image sticks around until your iPhone can take it. We tested this with multiple screenshots, and they all stayed queued up until our test iPhone was turned back on.
We should also note that screenshots taken on the Apple Watch are much lower resolution images than those taken on iPhones, iPads, or Retina Macs. The Apple Watch has a very high resolution display for its size, but the total pixel count is still quite small when compared to larger devices. Accordingly, they will appear smaller or of inferior quality when viewed, especially on your iPhone or iPad.