In a nutshell, a screencast is a video recording of whatever is happening on your computer screen at a specific time, often with audio narration recorded live too.
Screencasting has so much potential in teaching and learning! When I get a minute I'll share some examples demonstrating how creative you can get with screencasts.
For now then I've just embedded a screencast I did using Screenr a while back during an 'e-Learning clinic' at College where I was helping a colleague work out how to do gap fills in SMART Notebook.
I think Screenr is currently by far the best free screencasting tool out there: it's very simple to use; provided you have Java on your computer there's no software to install; the resultant videos are already on-line so you can just link to them or embed them into web pages without having to think about file sizes and video formats.
And here's a screencast of a screencast (!) by the marvellous Dave Foord explaining how to do a screencast in screenr. The recording is a couple of years old but screenr hasn't changed that much in the interim.