Although now my usage of Firefox is in sharp decline, thanks to the thousands of extensions available, there is always a moment when it becomes necessary.

These ones are the most useful, in my opinion:

Generic:

• Clean and close - Allows you to delete completed downloads and close the Downloads window with a single click
• Deng Google Bookmarks - Inserts a bookmark menu to access Google Bookmarks. Definitely useful if you use more than one computer and don't like Firefox Sync
• DownThemAll - The best download manager for Firefox!
• Lazarus - A must! In the event that the browser crashes or the web page has a timeout error while you're filling out a form, you can open it and fill it again with one click! Cool!
• IE Tab - Allows you to open the page with one click using Internet Explorer. The original link does not work anymore in the latest Firefox (5 versions in the last year!), There are several clones, one is an adware, but the best one is IE Tab 2
• Rikaichan - A great extension that allows us to know the meaning and the reading of Japanese words unknown to us (Also install the dictionaries - You need to know the hiragana)
• Super Start - A start page inspired by Google Chrome

For advanced users avanzati:

• Aging Tabs - Change the color of the less used tabs, it is very nice, too bad it does not work in the latest versions of Firefox...
• Ghostery - Blocks tracking scripts
• Gmail Manager - If you have many, too many gmail accounts, by using this extension you can manage them all without too many worries
• Greasemonkey - Allows you to change the behavior / appearance of some websites by downloading the appropriate script from userscripts.org
• NoScript - Blocks all the javascripts except those authorized by you, for added security - Use only if you are paranoid - many sites will not function as well as before
• Springpad - For adding notes to any page - terribly useful and productive
• WiseStamp - For adding a custom signature to Gmail emails

For web designers:

• Colorzilla - For seeing the color of any object on the page
• FireFTP - One of the best FTP client I've ever tried! The integration with Firefox is great!
• Firebug - All-in-one tool for development
• Measureit - For measuring the elements of a page in pixels, is more useful than you can think
• Screenshot Pimp - To make a screenshot of an entire page, even the parts that aren't visible. WARNING - It's a very invasive adware - install only when necessary and then immediately remove

Skin:

• Qute - A very balanced skin that adapts to each operating system. Too bad that is no longer compatible with new versions of Firefox ...