Can I run multiple copies of Terraria if I bought it on Steam?

A good way to do this is to run a Terraria dedicated server. This is a separate executable that runs the server in the background, leaving you free to launch Terraria and connect to the server, or play single player while the server continues running for everyone else. To run the dedicated server, browse the Terraria folder in your Steam folder, and run TerrariaServer.exe. For further information, and to set up port forwarding if required, visit the Dedicated Server page on the Terraria wiki.

If you're on a local network you can start Terraria client on one computer, yank the internet for just that computer, login to Steam on the second and start the client again there, and yank the internet again for that computer. Restart Steam in offline mode and reconnect the internet on both computers.

It would probably pay to buy a second copy of the game itself on a separate Steam account, but this could function if you're in a pinch.

Insofar as running the server, I recommend TShock. It's a bit more beefed up than the regular server software (commands, user groups, login passwords, all that good stuff) but accomplishes the same.