Annoyance Removal

AdBlock 0.5.x
Allows you to filter out any source URL on a web page—ad images, embedded frames, scripts, and so on.  You train it to weed out things that annoy you, and then you never see them again.  This is a must-have.
AdBlock Filterset.G Updater
This automatically updates a standard AdBlock filterset.  If you don't want to spend a lot of time training AdBlock, this is a great extra tool to have.
BugMeNot
Note that this extension is not on the Firefox page.  BugMeNot maintains a collection of publicly shared account names and passwords for sites that do (but should not) require them, like most online news sources these days.  You can use the BugMeNot site to find these things manually, but the extension makes it work automatically in Firefox.  May I suggest Miss Manners's column on the WaPo website after you install the extension?
Disable Targets For Downloads
Don't you hate when blank windows open up because you clicked on a download link?  This makes that not happen.
Linkification
It sucks when people put perfectly good URLs in a web page and don't make them active.  Linkification does it for you, turning plain old text into hyperlinks as appropriate.
Nuke Anything Enhanced
After AdBlock, perhaps the greatest extension of them all.  Annoying crap on a web page you are trying to look at?  Right-click, select Remove this object, and it goes away (at least until the page reloads).  I use this all the time.  Some sites you won't visit often enough to bother training AdBlock, but you can Nuke stuff anyway with this!
Print Preview
So, this is a pretty minor extension, but have you ever wondered why Firefox doesn't give you a bleedin' Print Preview button on your toolbar?  This extensions does.  (Oh, you do know you can right-click on the toolbar to Customize it with whatever buttons you want, wherever you put them, right?  Right?)
Redirect Remover
When you click a link, you want a nice clean visit to what is supposed to be the target URL, yes?  Not that URL embedded in somebody's frame; not that URL in a new window, unless you tell it to; not clicking through three pages warning you that you are about to leave somebody else's website.  This strips redirections down to the bare-naked unadulterated link that should have been there in the first place.
Sort Extensions and Themes
How do you know which Firefox Extensions you have?  You bring up the Extension Dialog, of course…and then you wade through it because it is kept in FILO order.  What were they thinking???  This extension keeps the rest of your extension alphabetically sorted, as they bloody well should have been to start with.

Information

ForecastFox
While this seems just to be window dressing, putting a couple of small icons with weather reports into the bottom toolbar of the browser, I use it a lot.  I expect anybody who regularly uses a customized web portal, like Yahoo! or Google, won't need this extension, but as I don't it seems quite handy.
GDirections
This is very neat.  Highlight text in a web page, right click, and then send it straight off to Google Maps for mapping or directions.

Search

Better Search
This embeds page images next to search results in Google, & al.  I think this makes it easier to scan a search result list for the sites you should be visiting.  Obviously, you do not want this extension if you are not using a broadband connection.
Search Engines
You have probably noticed the Google search box up on your Firefox toolbar.  In case you didn't know, you can get to it quickly with the CTRL-K keyboard shortcut.  Also, if you visit the Add-ons page, you can put additional search engines in there.  You will probably find ones (IMDB, Food Network, Dictionary, Wikipedia, &c.) that you will end up using every day.

Utility

GMail Space
This lets you use your GMail storage as a network drive, turning Firefox into a file transfer app.  2GB of storage wherever you need it can be pretty useful.
IE Tab
Sometimes you have no choice but to run Internet Exploder, either because some dumb-ass doesn't know how to write a web page, or because you are trying to be nice and develop your pages even for people who use half-baked, bug-riddled POS browsers.  Use this extension and turn a Firefox tab into an embedded IE instance just like that.
Reveal
Okay, this is just eye candy, but it is very nice eye candy.  If you have lots of Firefox tabs open at once, you might like this.  At the touch of a button it gives you images of every active tab for a navigation shortcut, and you can search all open tabs concurrently while you are at it.  It gives you a nice quick tutorial when you first run it; that ought to be enough to figure out whether you like it or not.

Web Development

ColorZilla
This adds a color picker app to your bottom toolbar.  That's nice, in and of itself, if you are matching colors to graphics as you develop a web page.  It includes some other goodies, like highlighting elements as you hover over them when ColorZilla is active.  You can copy selected color values (in most any format) into the clipboard as you use it.
DOM Inspector
Lets you view the DOM tree for a document.  Used in conjunction with View Formatted Source (below), it lets you see context specific style cascades at work in a target document.  This is great for debugging CSS issues during development.
Edit CSS
Brings up all currently applied CSS styling in the sidebar and lets you edit it live.  I abuse this—if I am reading ragged justified text, like a long news article, I force the page to show it justified.  Hey, whatever works, right?
View Formatted Source
A must-have if you develop web pages.  It lets you look at nicely formatted source in collapsible chunks.  It will show you source for a highlighted selection.  It will identify and tag block elements in the document itself.  Used with DOM Inspector, you can actually view the CSS cascade on every element.  Don't leave home (or develop web pages) without it.
Web Developer
This is a kitchen sink of useful features for web page development—turn things like JavaScript, caching, images, and cookies on and off at will; look up detailed information about the current page; ship the page off to assorted validators; and play with forms as you build them.  It gives you a clunky toolbar that you will want to have hidden most of the time, but it can be quite useful when you are in build mode.