Can I move photoshop elements from my laptop to a desktop computer?
Yes you can. you are normally allowed to install the software on two machines but if you have already done this then first deactivate the product from Laptop by going to:
Help >> Deactivate
Now this should make your life easier to activate on your desktop.
You can install it on another computer but you cannot move it. PSE is a very complex program with bits and pieces all over your hard drive, so you need to install from scratch every time.