A few weeks ago I changed my web browser from Internet Explorer to Firefox Mozilla. I have Mozilla set as my default browser, but whenever I log onto the web, it will always use Internet Explorer. Also, I have noticed that some items that I have in my Favorites Folder are listed as Internet Explorer and others are Mozilla.

What do I need to do to have Mozilla as my browser in every situation?

Not sure what you mean by "log on to the web" but it might be that your ISP requires the use of IE to do the log in. The IE favourites are those you imported, and the Mozilla are those you created/used with FF.
Cheers,
John

By the way, there is an extension for FF (View in IE) that will allow you to click on a link (or click on a page) and view it usinging IExpress. This would allow you, hopefully, to launch FF and then proceed with your login using IE.

Here's something to check:

IE / Tools / Internet Options / Programs tab
Check, at the bottom, to see if "Internet Explorer should check to see whether it is the default browser" is unchecked.