Firefox (or Mozilla Firefox) is an open-source browser which serves as an alternative to Internet Explorer. It is completely free to use, but you're also allowed to change the code should you want that. It is based on Netscape code, but developed by different people. If you didn't like Netscape, don't let this throw you; the only thing actually similar now is the way they process HTML. Most everything else about it is different.
The reason you will hear most often is SH/SC is that Internet Explorer is downright insecure and vulnerable to spyware infections. In fact, the Department of Homeland Security recently recommended people switch to another browser due to known security holes in IE, primarily revolving around VBScript and ActiveX - neither of which Firefox supports, for this very reason.
Security concerns are far from the only reason to consider switching to Firefox. Casual users and power users alike will be able to enjoy a wealth of new features:
Tabs. Tabs. Tabs.
Do you search a lot? Find As You Type is a quick way to search for any term on a given page. Forget all about Ctrl-F. Simply start typing your query and it'll automatically jump to the URLs containing the string. The functionality can be changed so that the searches include everything on a page.
Keywords will allow you to simply type "sa" in the address bar and the browser will jump to www.somethingawful.com for you. You can also set it up so that typing "sa icq prank" would open the somethingawful.com article search and show you any results with the words "icq prank" in them.
A built-in popup and cookie blocker.
Web developers (even IE-exclusive ones) will find that the tools provided with Firefox are topnotch and will help them troubleshoot javascript errors or annoying rendering quirks.
Tweakers or minimalists will enjoy the immense configurability that Firefox offers. Want to remove a menu? Fine. Want to reduce the UI to a minimum? Okay. Want to rid yourself of Flash ads? Also possible.
Roachfiend's Collection - Including GoonMenu and BugMeNot, these are the extensions roachfiend has designed for your enjoyment.
Text to Image - Would you like to automatically replace links to images by the images themselves? If so, this extension is for you.
Extensions: General
Tabbrowser Preferences - This one lets you have a lot more control over where links open. Essential in order to really take advantage of tabbed browsing. Comes in two varieties.
AdBlock - Unless you have an ethical problem with blocking ads, you'll want this extension. See below for a link to pre-made filters.
IE View - This is for those pages that abuse JavaScript or ActiveX, and therefore only show up right in IE - all it takes is a right-click and "View in IE".
Web Developer Toolbar - Do you do webpage design? Then this extension will simplify your life tenfold, if not more. On the fly CSS editing, two-click validation, and dozens more functions.
Radial Context Menu - If you are a fan of mouse gestures, this extension is for you.
Show Failed URL - So good it should come with the default installs from Mozilla. This extension will keep the URL shown in the address bar from being replaced by gibberish upon hitting a connection error. Use it together with browser.xul.error_pages.enabled to give you decent error messages that don't use dialog boxes. See the site for instructions.
Links
Firefox Help - Firefox Help should be your #1 resource for all things Firefox. The themes and extension part of the site is handed over to Mozilla.org, but it still contains a vast wealth of information and tips.
MozillaZine forums - If you have a problem with Firefox, chances are that somebody else has already posted about it here and possibly gotten help. Well worth visiting since it's pretty much the Mozilla Firefox community site.
0.9+ Compatible extensions list - You'll find a few extensions (such as the ones linked on the pornzilla page) that are meant for Firefox 0.8 and won't work with 0.9+. Look on this page in that case, they're almost all on there.
How to write extensions - So you've decided to write an extension yourself? Try roachfiend's tutorial.
http://mycroft.mozdev.org - Repository of extra search engines for the search toolbar (to the right of the address bar). From FreeDB to Amazon to alt.binaries.nl.
Firefox Tips - A wiki article dedicated to tweaks which get asked for often.