The standard license for Mathematica covers one computer. If you get premier service, you get a second 'home use' license, among other goodies. Each installation has a separate activation key, and it is only good for one CPU. You cannot use the same activation key on two different computers.
This is for the desktop. I have Premier service plus, which includes Mathematica On line. As far as I can tell, I can use this with any computer with an appropriate web-browser, although I have not tested it with simultaneous connections.
If you want to do heavy computation, you will want the desktop version, of course.
If you did not need to use both computers at the same time, it might appear that you could simply transfer the license to the computer that you are using. However, the way this is accomplished at the moment is a non-trivial task, involving an e-mail (or phone call) to Wolfram for a system transfer. I only do the when I get a new computer, and it can take a day.