I haven't used AOL for years when it was something like AOL5, so forgive me if this is a load of baloney.
If you can export your favorites from AOL and save them, they may save as HTML, if that's the case you will be able to import them into Firefox.
When you say open it from the desktop I take it you mean double clicking on the desktop icon. If so try right mouse clicking on the icon and checking in properties that the icon path is correctly pointing to the EXE file to open Firefox.
You never used to be able to use anything else apart from AOL at one time, but things are a changing and reading some of the other posts you can use IE, but I don't know about other browsers.