Prospective Beginner: HDD/OS Question

I'm thinking about hopping on the cheap-built PC idea. (I've never built one before).

My question is: If I transfer the HDD from my current computer to the new PC, will Windows (7) still function?  I just wanted to know if I could do this to save on the cost of buying a new copy.

You should be able to. You'll have to uninstall all of your old drivers (I would recommend installing Driver Sweeper, running your computer in safe mode, and erasing all the drivers with it before you put it in your new computer), install the new drivers for your motherboard; GPU; etc, and you'll probably have to re-activate Windows because it will detect that your hardware has changed.