Preparations:

You'll first need your IP address with whom you're visible to the internet.
The WAN IP address your modem/router gets assigned by your ISP - you can check it with www.WhatIsMyIp.com for example.
That's the IP address people will have to enter to join your server when it's running.

Also log in to your router and check if the Port 7777 (Terraria standard setting for it's multiplayer, if you don't change it) is forwarded to the outside internet world.

Setting up the dedicated Terraria Server manually:

Find "TerrariaServer.exe" in the Terraria installation folder and run it.
A black command promt window with white text should have opened.
This is the server control interface, where you will be doing pretty much everything (entering commands and viewing information).
When it starts, it should ask you to choose a world.
If you have a "clean installation" (you just installed Terraria and did not play it yet), it will probably have no worlds.
You can create one server-side by choosing the "n"-option given.
Otherwise it will list the worlds currently available on your computer. The last two command options create a new world, or delete an existing one.
Follow the on screen menu and your dedicated server is running.

Test it by starting Terraria and connect to it (if testing on the same computer where the server is running, go to Multiplayer->Join->Select your character->Type in "localhost" or "127.0.0.1".
If you are on a different computer, type in the IP address of the computer running the Terraria server.

That's how it's made ;-)

Have fun...

Kind Regards

GearsOfSera