Jing and ScreenToaster are great screen-capturing tools, but I find them pretty limiting, because you can't edit the video. Well, those days are over.
GoView (in Beta) is a free tool that allows you to record your screen with audio, then edit the resulting video. While the editing is pretty basic, you can remove the unwanted parts, and add titles. Perfect for student tutorials.
Unfortunately, the video is all online, hosted on the GoView site, so you won't end up with a file you can use, but it's easy enough to share the link.
Speaking of which, here's a link to their tutorial.