This isn't an SQLYog specific question, it's probably more general to software licensing. I'm probably changing my computer to a new one in the next six months, but I'd like to have SQLyog enterprise on my current one. Can I transfer a licence to a new computer if I'm just tossing the old one in the garbage or do I have to buy the program all over again?

We do not use any kind kind of 'activation' model nor locks the registration details to specific hardware in other ways.

If you have a binary/installer and corresponding registration details you can register on another computer as you like - but we request of course that you only have/use the # of installations as per your license!.