Screencast recording on a Mac.
The simplest way to create a screencast on a Mac is to use Quicktime. Apple offers step-by-step directions for recording a screencast through Quicktime. A video of the process is embedded below.
Screencast recording in Windows.
I use Screencast-o-matic to record on my Windows computers. There is a free browser-based version of Screencast-o-matic and a paid desktop version ($15/year). The free version is great for most situations. The desktop version offers some editing tools and longer recording times.
Windows 10 users have an option to create screencast videos if they install the Xbox app. The following video explains this process along with some of the shortcomings of the process.
Screencast recording on Chromebooks.
TechSmith offers Snagit for Chrome which supports creating screencasts that you can save into your Google Drive account. To use the screencasting option in Snagit for Chrome you will have enable the both the Snagit for Chrome extension and the corresponding Snagit Chrome app. The Snagit Chrome extension is what allows you to capture your screen. The Snagit Chrome app allows you to save your screen captures in your Google Drive account. You do have to install both the extension and the app for Snagit to work correctly.
A Google+ Hangout On Air allows you to broadcast your screen (you can do this even if no one else is in your Hangout). The broadcast is automatically recorded and saved to your YouTube account. Brian Bennett posted detailed directions on the process here.
My recommendation for a browser-based screencasting tool is Screencast-o-matic's free version.
There is no shortage of apps that will let you create whiteboard videos in which you draw and talk. But recording yourself demonstrating how to use an app or how to complete a workflow process on an iPad isn't so straight-forward. The best option that I've used isn't free. That option is to use AirServer's recording tool. AirServer is available to schools for $8-$12 depending upon volume.
This is the most difficult of all screencasts to record. Again, there are apps for making whiteboard videos, but to record actual app usage or workflow requires a number of steps. Labnol offers detailed directions on how to create a screencast video on an Android device.