Please note that recording may slow your device down if your cpu is weak, and requires a lot of storage to store the video files. Recording a game may slow down the game also, depending on your device.
Android Screen Recording Applications
SCR Pro is my favourite screen recorder. It requires root access. There is also a version for on Android 5.0 Lollipop. If you are on Lollipop, it only requires root access if you want to record the internal audio. I have found on my NVIDIA Shield I can record games with little to no lag. Here is a video I recorded of Asphalt 8 demonstrating that:
There is also an add-on that lets you broadcast to Twitch.
SCR Pro and the Twitch broadcasting add-on are paid apps.
If you want a free application that does the same thing, check out Shou.
Shou.TV is a free application that requires root unless you are on Android 5.0+. Again, if you are on Android 5.0 you will only need root to record internal audio. Shou.TV includes a broadcasting feature that lets you broadcast to the website of the same name. You can also choose to record the screen only if you don't want to broadcast to the world.
That's all for now, stay tuned for more posts!