Firefox and Internet explorer are both web browsers. However, there are many others.
However, the four largest browsers are Internet Explorer (from microsoft), Firefox (from Mozilla), Opera (from a company by the same name), and Chrome (from Google)
Internet explorer can respond quickly, and has a relatively low memory footprint(good), however, it is filled with security holes, and leaves your computer vulnerable. Internet explorer uses the Trident rendering engine.
Firefox has a large memory footprint (bad), however, it allows for a large amount of customization via add-ons, and skins. Firefox uses the Gecko rendering engine.
Opera has a low memory footprint and responds somewhat quickly, but is difficult to customize. Opera uses the Presto rendering engine.
Chrome has a medium memory footprint, and responds fast. However, given that it is relatively new, it is hard to customize, though this will be fixed once Chrome establishes itself more. Chrome uses the WebKit rendering engine.