What is "source code"?
In the same way that food has a recipe, humans have DNA, and buildings have blueprints, software has source code. It looks like gibberish, but when run through a compiler, out comes a software application. If anyone gets their hands on the source code, they can duplicate and edit that software at will. (Like having the recipe to McDonald's secret sauce, I would be able to make as much of it as I want, without going to McDonalds) The source code of Netscape Communicator is called Mozilla.
In 1998, Netscape released the source code of Netscape Communicator to the public. Why? It's a different method of software development called "open-source". By releasing the source code, programmers everywhere will be able to download and edit the code at will. They can submit that edit back to the project website, with hopes that it will become a part of the product. The name given to the open-source development project for the Netscape Communicator code was Mozilla. The organization that co-ordinates the development project is called the Mozilla Organization.
When the Mozilla project began, much of the feedback was that the code itself was too complicated to learn and contribute to. As a result, there was little in contributions from outside Netscape employees. It was decided that it would be best to change the layout engine, which constituted a total rewrite of the code. The new layout (Gecko) was cleaner and much easier to learn, thus contributions would increase. Netscape Communicator was laid to rest, and Mozilla.org had a completely separate product.
Netscape employed programmers to work full-time on the Mozilla code, as did other companies such as IBM, Sun Microsystems and Red Hat. The mandate of the new code was for better performance, portability and standards compliance.
With there being new code in development, it must be tested in a variety of cases. Binaries are released by Mozilla.org with the intent that those, who are not programmers, can still test the code and give feedback (report bugs, suggest features, etc.) Of course, with this being new code, it had to be given it's own release numbering system. At first they were called milestones; then there was a manifesto set for Mozilla 1.0, so the numbering system had actual version numbers.
A total rewrite of the code takes time, and by the year 2000, Netscape hadn't released a browser based on the code developed at Mozilla.org. By this time, web standards had evolved and Communicator did not have the rendering capabilities to keep up. The Netscape marketing team decided that the Mozilla code was stable enough to release as a Netscape branded end-user product; and even though Mozilla was still over a year away from 1.0, Netscape 6 was released using the code from Mozilla 0.6.
With Netscape being a vendor of the Mozilla code, they can customize it how they want, as well as add additional components developed through closed source development paid for by Netscape. The Netscape distributions include the Netscape brand name, Netscape default settings, it's very own spellchecker, an integrated Instant Messenger application, and integration with Netscape.net webmail. Netscape also has licence to distribute plugins with their product, and the resources to provide technical support for its users.
Why did people use Mozilla?
As much as Netscape offered many extras on top of the Mozilla code, there were many people that preferred Mozilla releases. The Netscape 6 series lacked many features from the old Netscape Communicator code, causing Netscape users to keep a closer eye on Mozilla development. Netscape did not release a new version with every Mozilla release. There was a ten month gap between Netscape 6.2 and 7.0, in which Mozilla released ten new versions. Many users also did not like the mandatory AOL additions that came with Netscape, and preferred the non-branded code. Even though Mozilla made it quite clear that the binaries were intended for testing purposes only, nobody really minded that people were using the code. It also ended in an increased amount of feedback.
The Mozilla Foundation:
On July 15th, 2003, AOL ceased development of Netscape, and let Mozilla go independent. Mozilla was already being treated as an end-user product by many people, and more migration was expected of Netscape users. This prompted Mozilla to officially start marketing toward end-users, and providing technical support.
The New Netscape:
The team running the Netscape.com portal were not very happy with AOL's decision to discontinue Netscape, since a good portion of their traffic is from clickthroughs using the Netscape software. As a result, they took it upon themselves to contract former Netscape developers to make 7.2, using the money from advertising space on netscape.com, thus bypassing the AOL product development team.
What about Firefox and Thunderbird?
The Differences between Mozilla versus Firefox and Thunderbird