I use Firefox mainly because of the extensions which are available. Here is a list of the extensions which I find indispensable. The list is roughly sorted in terms of importance. The links all go to the official Firefox Add-ons site whenever possible. There you can find more information, reviews and a link to the extension's home page.
- Adblock Plus because graphical ads are too annoying
- CustomizeGoogle because it makes Google more convenient to use and allows me to easily search on other search engines
- Add Bookmark Here 2 because it makes bookmarking easier
- Paste and Go so I don't have to separately paste and then hit go or search
- Resurrect Pages so I can easily get to a cached copy when a site is down
- BugMeNot because registration is annoying
- dragdropupload because it makes uploading files a lot easier.
- Permit Cookies so I can easily see if I'm permitting cookies for a site and change cookie permissions
- FlashGot for lightweight integration with my download manager
- RefControl because the Referer header doesn't do anything useful for me
- Tab Clicking Options because remove buttons from tabs and use double-click to close tabs
- FxIF 0.2.2 so I can see EXIF data from online photos
- CuteMenus - Crystal SVG because it makes Firefox look nicer and sometimes makes menu items easier to find
- QuickRestart so I can easily restart if I install, uninstall, enable or disable an extension
I regularly try out new extensions. Here is a list of extensions which seem promising. I haven't used any of these enough to strongly recommend them, but I do think they're at least worth checking out.
- Second Search because it's a convenient way to use alternate search engines via the search bar
- Adblock Plus: Element Hiding Helper 1.0 because it helps remove some ads, but these are fairly rare and inoffensive so it may not be worth it.
- Locationbar² because it allows easy access to parent directories on servers
- Redirect Remover because redirects waste time and bandwidth. Sometimes they're a good idea though, and those need to be entered as exceptions.
This list was created with the help of Extension List Dumper.