Chrome: Recording video of your desktop can become complicated pretty quickly. If all you need to record is inside your browser, Screencastify makes it a straightforward affair.
The extension records anything that's happening within your browser tab. You can choose whether or not to record microphone audio and you can even set it up to include a feed of your webcam in the corner of the video. There's also an experimental feature that can record your whole desktop, but to do this you'll need to set "Enable screen capture support in getUserMedia()." to "Enabled" in chrome://flags/.