The ability to take a screenshot on your Android device is one of those things that has been around forever, but became a much bigger deal once Google made it a standard feature that could be done right from the device instead of requiring a connection to a computer. The same could be said of recording a video of what is happening on your device, if Google ever gets around to making it a standard in the operating system.
In the mean time, you have two options that don't involve connecting to your PC: use a third party app that hijacks the Chromecast to Screen function and records that, or use Screencast natively in Cyanogen OS.
As the name suggests, Cyanogen OS Screencast is only available on devices running Cyanogen's flavor of Android. Accessing the feature couldn't be more simple — you just open the app and tap start recording. You'll see an icon in the top left to let you know a recording has started, and you can go and do whatever you wanted to record on your phone. Once you've finished, you can stop the recording by pulling down the notification tray and tapping on the Screencast notification. The recording will stop, and the notification will turn into a share action so you can send that recording wherever you want. There's a good chance you'll want to do some brief editing before sending it away to a social network, but the resulting video file is now yours to do with as you please.
It's simple, straightforward, and doesn't fill your notification tray with multiple icons like most of the third party solutions. An integrated solution like this makes a lot of sense even if users only end up using it to report a problem, but chances are the more users see features like this the more likely we are to see new uses for it.