Mozilla is the name of the platform and organization powering the Firefox browser and many other applications. Mozilla as an organization is an idealistic, highly professional institution, with the stated goal of "making the web a better place". They (we) have already largely succeeded at doing that.
Using Mozilla as a programming platform it's possible to construct web-enabled desktop applications that run on any operating system. Some examples are:
As a platform it's the one of choice for web development because of its standards support, security and openness - you can discuss policies, bugs and everything else with a huge community that is involved in the project(s). As a developer you're able to contribute code to fix and enhance the finished product.
It's possible to create extensions to enhance/modify the capabilities of Mozilla applications. For an example have a look at one I made myself: SMS Sidebar.
If you're not using Firefox right now there is really no reason to continue using MS Internet Explorer. Be at the forefront of innovation. Make the change.
Mozilla is one of the few companys that are able to cleanly implement and improve open standards
and also bring them to the masses. Unfortunately Microsoft is not, they have other interests.
Mozilla and thus Firefox integrate many modern and extremely useful standards that push the
envelope of what the web is capable of.
Mozilla's CTO Brendan Eich is the driving force behind the extremely successful programming language JavaScript (JS). Without it the web would be like reading an (illustrated) book with forms, links and video. All real interactivity would be almost inexistent or quite unsafe.