Mozilla was pretty central to the web in the early 2000s, with the release of Firefox 1.0 which was the browser that displaced IE6 in the minds of developers and the browser with an NYT ad paid for by donations. It was the first truly mainstream browser with good HTML4/CSS support, tabs, add-ons, and themes and the first browser overall that had developer tools (with the Firebug add-on). Mozilla also pioneered the idea that you could refinance browser development by selling the top spot in your search box (and unfortunately, they never had another such profitable idea; also, they kinda sold all of us to Google).

They've lost a lot of influence on web standards and web development since, and with that they lost cachet with users. A big part of that was that Google stopped promoting Firefox when they released Chrome in 2008.