I think the most useful answer I can give is that *generally* any suitable executable in the "cgi-bin" directory can be a web application. How ever C source code is not an executable so will not work. You would need access to a compiler to be able to use C but only the more expensive web hosting packages will offer that on security grounds.

Now additional points:

1. The "cgi-bin" directory is configurable (but not usually by you). For security reasons it should be outside of the document root.
2. PHP code is normally in the document root not the cgi-bin directory.
3. When you buy a webhosting package you normally have an idea which languages you want to use and you should check the terms before you buy. But support for languages is definitely server specific and so you need to check.