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!