This is generally something that you'll have to expose yourself, so yes, it will be dependent on the language your system is written in (though often it's possible to write wrappers for other languages as well).
If, for example, you had a program written in C, for Windows, plugins would be written for your program as DLLs. At runtime, you would manually load these DLLs, and expose some interface to them. For example, the DLLs might expose a gimme_the_interface()
function which could accept a structure filled with function pointers. These function pointers would allow the DLL to make calls, register callbacks, etc.
If you were in C++, you would use the DLL system, except you would probably pass an object pointer instead of a struct, and the object would implement an interface which provided functionality (accomplishing the same thing as the struct, but less ugly). For Java, you would load class files on-demand instead of DLLs, but the basic idea would be the same.
In all cases, you'll need to define a standard interface between your code and the plugins, so that you can initialize the plugins, and so the plugins can interact with you.
P.S. If you'd like to see a good example of a C++ plugin system, check out the foobar2000 SDK. I haven't used it in quite a while, but it used to be really well done. I assume it still is.