Perhaps you have used Koush's Mirror app in the past. Well, you haven't used this one—it's a completely new app with a new listing in the Play Store. Mirror still lets you beam your phone or tablet display to other devices, but it can also record the screen. The difference, now it can work without root on Lollipop.
One of the new features introduced in Android Lollipop is an android.media.projection API that allows apps to capture the device's screen. Unlike KitKat 4.4, where you could achieve this through ADB and a USB cable (or on your phone directly with some root privileges), this new API works out of the box and opens the screen recording feature to non-root devices.
When we first reported on this option, only a couple of apps had made use of it.
You might recall that Android 4.4 added support for native screen recording, but it was a very developer-oriented tool that required you to plug the device into a computer and use ADB to control the capture. Some root-only apps came about that made use of that system sans USB cable, but Android 5.0 adds an API for screen recording that doesn't require root. We're still in the early days, but it looks like there's one less reason to root now.
Google added native screen recording functionality in Android 4.4, but it only works over ADB and there's no audio output. Third-party developers have been working on ways to expand the usability of the native functionality, but the root only app SCR might be the closest to implementing sound and video perfectly after the most recent update.
In the new version, SCR gains support for experimental recording of internal audio with the video.
Android 4.4 has a new screen recording function for developers, but as the CyanogenMod team has already demonstrated, it can be adapted for more general use. Well-known developer Koushik "Koush" Dutta has taken advantage of this for Mirror, a new screencasting app for KitKat that allows any rooted user to either record video directly on a device, or stream to an Apple Airplay-compatible receiver like Apple TV. No custom ROM required.
Android has had native support for user-taken screenshots since 4.0, and a few OEMs like Samsung have had supported the feature even before that. But until now, getting a reliable video recording of your device's screen has been a major pain, usually requiring some kind of root solution that doesn't work for all hardware. In KitKat, Google is doing away with that, allowing end users to record video directly from the screens of their devices.