I don't have anything extra installed on my computer. Whenever I try to include one local PHP file on another, nothing happens. My question is if it should be working, or if I have to have some kind of server for it to work?
Well, you obviously need a server for a server side language... Don't you think?
Have a look at PHP servers such as Apache, or install an all-in-one such as XAMPP or WAMP.
At the very least, you need to install PHP and run it from the command line.