You'll need to include a lot more detail here. In what capacity are you recording the screen? Locally, you can use something like ScreenFlow, which is a third-party app you have to download and install.

For a remote screen, there are several proprietary solutions that all require plugins.

Without third party software or plugins, you can do this (on bleeding-edge browsers only) with WebRTC. This is currently way too unstable for a production rollout. But it won't be for too much longer. (This is an area where I can help, if you're looking into in-browser/custom solutions.)

Your best bet is probably Google at this point, followed by actually using the various apps out there for the end you're looking for. See what works best for you.

Good luck!

I've been using the pro version of Screencast-o-matic (a whopping $15/year)

http://www.screencast-o-matic.com/

Works well.

As Jason mentioned there are a variety of options depending on your needs. One question that comes to my mind is "Do you need to do post recording editing?" ...and "What operating system are you using?"

I have a MacBook Pro and Office 365. I use QuickTime (included in Mac OS X) to record my screen and drop the file in OneDrive (you could also use DropBox, Google Drive, or any other cloud drive). This works well for videos that don't need post production editing. In a pinch, you can use iMovie to do post production editing with QuickTime movies. Though in my opinion it can be challenging to use at times.

Additionally I have a copy of Camtasia Studio ( https://www.techsmith.com/camtasia.html ) I use if I know I'm going to be doing considerable post production editing.