I am slightly confused on how software like this works. Is this compatible with all game making programs, or is it just a plugin to some other program?
Tiled is a standalone program. It is used to create map files that can then be used in your game engine to load up maps. I wouldn't necessarily say that it is compatible with "all" game making programs but a lot of engines have Tiled support out of the box. Programmers can write a Tiled Loader in their favorite programming language.