I play Terraria a bit with my brother and friends and I am always the one with the server, but sometimes while I am playing it gets a bit boring and I want to play on my own world and character (I do know about that whole duplicating cheat and am not a fan) at the same time. I was able to do this for a while and now I can't. Is there a setting that I accidentally turned on or just an annoying patch?
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.
The easiest way is to open the steam in the folder location by right clicking the steam application. Then, open the steamapps, then the common folder, then the terraria folder and find the terraria application (should have the terraria picture). Finally just double click as many times as you want and you will have as many terraria windows open as you want.