They can be installed on as many computers as you want. But they won't work without you signing in. The beauty of Steam is, you can have multiple computers with your game, but you can only be signed in to one computer at a time.
If you're going to get rid your old PC somehow, just keep your hard drive and plug it into your new computer as a slave. Create fresh shortcuts to the executables on the old drive, and see which ones work. Some games work like a champ, some won't. Steam games work fine (I've done this at least twice myself).
You may use your Steam account on any machine which can connect to the Steam network - Steam allows you to download and install any games registered to your account as soon as you log in.