Steam installation:
Look for tModLoader in the Steam Store and install.

Manual installation:
Make a 'tModLoader' folder next to the Terraria install folder and place the .zip contents in it. 
Launch the provided 'start-tModLoader.sh'/'start-tModLoader.bat' file.
For GoG users -> delete Steam_appid.txt to switch from Steam to GoG

Run tModLoader:
Just launch the provided tModLoader file (start-tModLoader.sh/start-tModLoader.bat) or launch through Steam. 
If you see Mods or Mod Browser on the main menu in Terraria, you have successfully installed tModLoader.

Uninstall/Switch back to un-modded Terraria:
In Steam, right click tModLoader in your library -> manage -> uninstall
If you installed manually, simply remove the tModLoader folder.
Note: If you want to play un-modded Terraria or join an un-modded Terraria server, you can't use tModLoader.

Dedicated tModLoader server:
If you are installing tModLoader onto a machine that will be running a dedicated server, 
you'll need to install the un-modded Terraria server first: 
http://terraria.org/server/terraria-server-1353.zip  
After installing, you can follow the manual install instructions to copy the tModLoader files into the folder you installed the server into. 
More info on running a dedicated server can be found at https://github.com/tModLoader/tModLoader/wiki/Command-Line#dedicated-host .