Ok, since you're not new to HTML, you should understand that if you go to a directory instead of a specific file, your web server is going to try to feed you one of the index files (index.htm, .html, .php, etc...). That's why there's the index.php file there.
On the first time you run the calendar (after you've completed the upload procedure), the index.php file will detect that it has not been configured and suggest you run the Setup Manager to do so. After the calendar is initially configured, all the index.php file does is redirect you to the demo.php file which is there as an example to display some of the capabilities of the calendar and to give you an example of how to integrate the calendar script into exsting web pages.
If there was no index.php file, depending on your webserver installation, your server might display a file listing or a 404 error... and neither is desirable.
On the other hand, you might not want users to be able to go to /calendar/ and view the calendar. That's why the option is in the Setup Manager to disable the loading of demo.php from index.php. So that if a user goes to /calendar/ all they get is a blank page (not a 404 or a directory listing). It's "prettier" and might be considered a security feature.
Now, if you link directly to demo.php, even if it's disabled in the Setup Manager, it will still display. But since the demo.php file is more of a demonstration of what the calendar can do, and an example of what a properly formated file looks like, demo.php can be deleted after you're satisfied with the way you've integrated the calendar (as long as you're not using it of course).
For integrating it inside some of your own pages (like you're saying "integrate a header") make sure you read the Getting Started Guide
and also check out the "Integration" section of the Online Documentation
A quick way to understand what to do with those instructions is basically, create your own page with all your headers, and menus and all that fun stuff that follows the rest of your site design. Then follow the integration guide to add the little bits of code where you should (like in the <head> section, then right after the <body>, and then finally exactly where you want to show the calendar).
If you run into problems, we're here for that and we'll try to answer in the shortest delays.
[Some text in this post has been modified by Brian]