There are several options for screencasting.
Two of the most developed, complete options are Camtasia Studio, for Mac and PC, and ScreenFlow for Mac. However, both these options have a moderate price tag, and might be overkill for simple use cases.
Screencast-O-Matic is a great, inexpensive, web-based screen recorder for Mac and PC, and it's the go-to choice for several of our staff.
CamStudio is free software available for Windows. It does not have extended features, but it get's the job done and works well.
QuickTime Player (which comes built-in on Macs, and is available for Windows here) is reportedly able to do screen recording, but features are limited. It is, however, free.
Finally, if you happen to run Ubuntu Linux, we recommend Kazam. Should also work on other (non-Ubuntu) Linux desktops.