Even though the mouse is plugged into the CPU, the curser isn't moving on the screen. The mouse was removed from an old desktop a couple of years ago to be used on another desktop and was working fine but now it isn't working at all. I have not used this desktop in about a year. Can this issue be helped at all?

Hi welcome to fixya
if you are using optical mouse clean first the optical sensor,dust can cause malfunction. now after you cleaned checked the wiring through continuity resistance if one of the wiring has loose connection the mouse will not works fine. now if still you can not use the mouse you bad circuit board just replaced with new mouse so you can use it to your desktop. hope this will help you thanks for consulting us.

First of all what kind of mouse is it? USB (metal retangle) or PS/2 (round plug with small pins)
USB is plug and play (meaning you can plug it in anytime even with the computer on and it should start working)
PS/2 The computer needs to be off before you plug it in.
Also some computers you will need to enter BIOS and enable legacy support for PS/2 mice/keyboards
To enter BIOS, while the computer is first booting up press the key it asks for to enter setup/settings, generally it is F10, F11 or del. Look for legacy support and make sure it is enabled. Save and Exit BIOS.
Let me know if you need further assistance