Just uninstall from the old machine, then install and activate on the new machine.
There's no uninstaller, so just delete the particleIllusion Application folder. You can move your Emitter Libraries folder from the old machine to the new one if you want -- you'll definitely want to do this if you've modified any libraries or created your own.
After successfully moving (and using) this program to my MacPro, I did not use it for a month or so. Today when I opened the program it told me I needed to activate it. In addition, while waiting on the "Activation Window", to find my serial number, the program just crashed.
Running OSX 10.6.5 - are there problems with that build of the OS?