Every time you go to a website, you send what is called a User Agent string. Every browser has one. Most of them start with the word "Mozilla". Mozilla has always been the nickname for the browser made by Netscape. (And now, we have a browser called Netscape and a different browser called Mozilla, both of which identify themselves as Mozilla).

Let's take a look at a few examples:
Internet Explorer sends this:
"Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1)"
Opera (by default) sends this:
"Mozilla/4.0 (compatible; MSIE 5.0; Windows XP) Opera 6.05 [en]"
Mozilla sends this:
"Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.1)"

They all start with the word "Mozilla" because an early version of Netscape (which identified itself as "Mozilla") could handle more complex HTML than early versions of Internet Explorer. Web designers were swept up into a practice that would haunt them yea unto the gates of hell: they started Sniffing Browsers. Basically, they looked at the User Agent string and started to give different content to different browsers. Then Internet Explorer added functionality to handle more complex HTML, but since some folks had already started looking for the word "Mozilla" in the User Agent string, Microsoft decided that they would adopt it too.