There might be an app in Google play to record your screen but it’s either paid, buggy or needs root access. I’m not ready to root my phone just yet and I am looking for a free alternative. The best option is to use the android debug bridge (ADB) from android SDK. Not that easy though but as long as you follow the steps from the video you should be good to go!
Download Android SDK here: http://developer.android.com/sdk/index.html
ADB commands cheat sheet:
adb shell screenrecord /sdcard/test.mp4 (record)
adb shell screenrecord –-bit-rate 8000000 /sdcard/test.mp4 (record in 8Mbps)
adb pull /sdcard/test.mp4 (download video to your computer)