How do I take a screenshot of the current window on my Lumia phone?

This depends on which Lumia Windows Phone version is currently installed on your phone, go to Setting > extras+info:

Lumia Black and Amber: Press and hold the Power and Start (at the bottom of the screen) keys at the same time.

Lumia Cyan and Denim: Press and hold the Power and Volume Up keys at the same time.

When done correctly, your handset screen will flash and play an audible camera-like shutter sound. The screenshot is saved to the folder in Albums/Screenshots, available via the Photos hub.