Question:
Can I use my Steam account on other computers?
Answer:
Yes, you can use your Steam account on as many different computers as you like - but only one at a time!
For example, you could use your account on your work computer during the day, and on your home computer at night. On the weekend, you could even use it at a friend's house. Steam will automatically download all of your games to whatever computer you are currently logged in on, and your Friends list will be automatically duplicated on that machine.
If you have multiple Steam accounts on one computer, they will all share the core set of downloaded games, but each user's game configurations and any custom content will have to be copied or transferred.
You just can't use a single Steam account on two or more computers simultaneously.
If you log into Steam with a second computer using the same account that is running on the first, the Steam ticket will be invalidated on the first computer and transfer to the second. The first machine will eventually be prompted for a login to establish a new SteamUserID ticket, and if the person using that computer logs in, the ticket will then be invalidated on the second computer and transferred back to him.
You need a unique Steam Account for each computer that is going to be online at the same time.
A similar case arises with Stand-Alone Steam Dedicated Servers, which also use Steam accounts.
If you are using a single Steam account for your Stand-Alone Dedicated Server and your client, you will run into the same ticket difficulties described in the previous paragraph. Client Listen Servers and Client Dedicated Servers use the SteamID ticket for the client on which they are run.
Auf gut deutsch: 2 PCs, JA! Auf einmal? Nur teils!