How can I move activated Unity from one computer to another and continue working on my project(s)? I mean, if I change my computer, or change i.e. motherboard/CPU. It might happen many times.
You get two activations for a single license. If you need one of them deactivated, send a mail to support@unity3d.com.
As Eric said, you can get more activations via the support e-mail. However, when this happened to me a few months ago, Unity Support told me that you can avoid this by uninstalling Unity3D off your computer before you install it on the new computer.
I haven't personally tried this yet, but I'd assume it's a solution for people waiting on support.