I am thinking of changing from a PC to a Mac. Will I need to buy a new, Apple compatible, version of Lightroom, or can I upgrade to it, or can I get it free as I've paid for the PC version?
Your licence is not platform specific, so Lightroom can be used on any two machines (Mac and/or WIndows), with one actually being used at any given time.
So no, you're money's safe!
What Keith has said, to be more specific you can install the same version of Lightroom for which you have a serial number.
If you wish to change the version say from LR 4 to LR 5 you will have to pay for the upgrade.