I mainly use the Firefox web browser (the successor of Mozilla, which I had used since April 2000) for the following reasons: free software, designed for standards compliance (HTML, CSS, PNG, MathML, SVG, etc.), portable (I can use it on various platforms), good from a security point of view (i.e. with relatively few security holes for its complexity, and they are quickly fixed), actively developped, with extension possibilities (see below).
Official Mozilla web site. And Bugzilla: the Mozilla bug database, to look for or report bugs.
MozillaZine Knowledge Base: documentation for Mozilla/Firefox, hints and tips...
MozillaZine: your source for Mozilla news and advocacy.
Mozilla projects (mozdev.org), and in particular the add-ons and the active projects.
For developers: Mozilla Developer Center.
Firefox extensions database. Here are a few extensions I use:
Lots of additional useful features for tab-browsing.
A site-navigation tool (formerly Link Toolbar). Eases navigation within sequences of pages by providing toolbar buttons for the first, previous, next, and last page. Also includes buttons to move up a level in the website, or to the top of the website, and a menu of links to further related pages (which can include a table of contents, an index, printable versions of the current page, translated versions, or page authors' contact details, etc.). Web pages can provide the information about the related pages using the HTML link element, or the extension can guess them [...]
.
Link Widgets does not work with Debian's Iceweasel, but can be patched.
Replaces Flash objects with a button you can click to view them.
Display the Google PageRank and Alexa ranking with search-related tools.
Link checker: looks for broken links on a page.
Customize the look of websites and of the user interface. In particular, I use this extension to fix the font size chosen by some web sites and to combine the Stop and Reload buttons.
Copies selected text to the clipboard automatically. Like Linux or mIrc. I use this extension under Mac OS X, with this patch (otherwise Auto Copy forgets to copy text that has been selected just before).
Adds a menu and a toolbar with various web developer tools. May also be useful for non-developers and for bug reports.
Provides a temporary browser frame below the main content-area. I find this extension useful to compare two web pages; in particular I use it to check the translations of my pages. Replaces the Content Holder extension.
Offers the possibility to display documents of unsupported types in the browser window; these documents may be simple text files, like C sources! Very useful extension, until bug 57342 is fixed.
Allows you to customize the way web pages look and function.
Gives information about the HTTP headers.
Generates an index of any web document structured with headings, indicating the level of the headings and its possible errors. By clicking on any of its items, you can access directly to the content of the page.
Documentation to build your own extension, and a tutorial by Paul Adams: Your First Firefox Extension (it's on an archive, so maybe a bit obsolete).
Like any software, Mozilla and Firefox have bugs and lack some features that would be very useful. Here are the most significant ones for my use of Mozilla/Firefox (some of them may be related to each other). Note that some of these bugs are present only in development versions (trunk), which are not aimed at the end user.