What Is Mozilla?
As far as most users are concerned, it's a (very) cross-platform* open-source web browser, with very good support of World Wide Web Consortium (W3C) standards.
*At the time of writing, Mozilla 1.0 is available for Windows 9x/NT/2000/XP, Mac OS 8.x and 9.x, Mac OS X, Linux/x86, Linux/Sparc, Linux/PowerPC, AIX 4.3.3, BeOS, BSD/OS/x86, BSD/OS/Sparc, FreeBSD/i386, HPUX 10.20/11.00, OpenVMS, OS/2/x86, Solaris/Sparc and Tru64 Unix. (phew!)
However, Mozilla is more than a web browser suite. It is also a cross-platform software toolkit, with particular support for internet client software. Almost as a side-effect, mozilla.org has also spawned an open-source bug tracking system, bugzilla.
The history
In early 1998, Netscape (the company) decided to release the source code for Netscape Navigator (then version 4.x), so that it could benefit from outside contributors when developing Netscape 5.
Certain parts of the code (e.g. the spellchecker) could not be released, since they were the copyrighted property of other parties being used under licence by Netscape, but the bulk of Navigator itself (layout engine, javascript, etc.) was released.
An open source group was formed, named Mozilla (after Netscape's Lizard Mascot) to develop an open-source web browser, based on this code, which would support as many W3C standards as possible.
Netscape are very prominent contributors (in terms of people and other resources) to the Mozilla project, but there is also input from IBM, Sun Microsystems, Red Hat, other companies, the academic world and private individuals.
A few months into the project, it was decided that the old Netscape layout code was unsuitable, at a design level, to the task of supporting such standards as Cascading Style Sheets.
This led to the Mozilla project discarding much of the old Netscape Navigator code in late 1998 and starting, from scratch, to write a new layout engine, dubbed Gecko.
The first major release based on the mozilla.org codebase was Netscape 6.0, which was released in November 2000. Most of the programmers agreed with the press and many users: the release was too early, and Netscape 6.0 was not a success. Netscape made further releases based on later revisions of the mozilla.org code, but the damage was done.
The next major release was Mozilla 1.0, in June 2002. While this still has quite a few rough edges, it was the first release that mozilla.org was prepared to call "finished". It was followed by the release of Netscape 7.0.