Rick...
Your home page is nothing more or less than the first page that opens up when your browser launches. You can set it to any page you want; from MSN to CNN to FOX News (or maybe not Fox News for you... perhaps something more left-leaning... oops! did I just let a barb out?)J/K... You can even set Cbid as your home page if that turns your crank...so to speak.
Your browser is the program you use to connect with and display pages on websites you visit. If you run a version of Windows, then Internet Explorer is your default browser "out of the box".
Mozilla is the name for the current incarnation of the Netscape Navigator browser that was popular a number of years ago. It was and is developed by a user group whose home on the web is www.mozilla.org you can go there and download several versions of the browser, including Firefox, which is the latest and greatest (supposedly) from the Mozilla folks. There are also "plug-ins" you can download and install to add a large number of custom features to your surfing experience. I use the Mozilla suite myself. I downloaded Firefox and ran it for a while, but for a number of personal-preference-type reasons, went back to using Mozilla, which is the older version. I do, however, use Thunderbird, the stand-alone email client software from Mozilla. All are free downloads, the install is simple and painless, and Mozilla seems to run and load pages much faster for me (I use a satellite-based internet connection, Starband) than IE. Many informed opinions say that Mozilla is a safer browser to use, as most hackers and virus perpetrators write their poison for the IE browser and email client, as there are more computers out there using it than any other and they want maximum damage effect, so they target the most popular browser. I don't know how true that may be; I am not a silicon-based life form. My world outside the computer is higher priority for me and I have not taken the time to become a techno-geek.
You can certainly get the Mozilla browser and try it, see if you like it... if not you can uninstall it and go back to IE if you like, since you don't uninstall IE in order to run Mozilla.
Good luck!
badger