Microsoft added the ability to take a Windows Phone screenshot in Windows Phone 8, and it's worked consistently ever since. But with Windows Phone moving towards a more integrated platform with Windows, the way you take a screenshot has now changed in Windows Phone 8.1. Why? To make it more consistent with Windows 8.1, of course.
OK, not really. (In Windows 8.1, you take a screenshot with WINKEY + VOLUME DOWN on a tablet.)
In Windows Phone 8, you can take a screenshot by pressing the Start and Power buttons simultaneously. Your handset screen will flash and play an audible camera-like shutter sound, a "Saving to Screenshots" banner notification will appear, and the PNG-formatted screenshot is saved to the Screenshots album.
When you do this with Windows Phone 8.1, however, you see an error message that explains the new button combination you must press.
As noted, you must now press the Power and Volume Up buttons simultaneously to take a screenshot. And yes, this is the same hardware button combination that works on Windows 8.1 devices as well. You can find the resulting screenshots in the Photos app: Just navigate to Albums, Screenshots.