If you are new to Linux you may read my guide to Firefox in Ubuntu, which provides some info about installing add-ons in Firefox.
End-User Add-onsAdBlock Plus | Install from Firefox Add-ons | Homepage
This add-on is one of the most popular extensions out there. It is probably one of the first add-ons users install after a clean Firefox setup. AdBlock Plus blocks all annoying, intrusive ads. It has an option to allow or not non-intrusive ads, and it can be enabled or disabled on specific pages on the user's request. It installs a submenu in the Tools menu and also appears on the add-on bar, at the bottom of Firefox.
AdBlock Plus in Firefox 19.0:
Video DownloadHelper | Install from Firefox Add-ons | Homepage
Video DownloadHelper allows you to download videos from various sites, including YouTube. It provides a configuration window, as well as an icon in the location bar for fast access to downloading options. It detects the videos embedded in the current web page and provides you with a list. The Quick Download option makes it easy and straightforward to quickly download any video in the active window. Files downloaded this way can be found inside the ~/dwhelper directory.
Speed Dial | Install from Firefox Add-ons | Homepage
This add-on provides a functionality which is similar to Opera's speed dial, that is, it shows thumbnail previews for websites of choice when opening an empty tab. Firefox already implements a speed dial page, but it shows only pages recently accessed, so if your history is empty the default Firefox speed dial will be empty. Speed Dial provides enough configuration options (via the Tools->Add-ons tab) to make it worthwhile. You should check the Load in blank new tabs option to enable it when opening empty tabs. There is a rather big number of configuration options, making this add-on extend this functionality in pretty much any way the user wants. It provides organized groups, can change the number of rows and columns, can change the update interval, can change thumbnails size, and includes keyboard shortcuts.
Speed Dial is feature-rich and enhances a new empty tab or window:
Screengrab | Install from Firefox Add-ons
Screengrab saves web pages as images, in either PNG or JPG format. It is simple and fast for those of you who like this feature.
Pocket | Install from Firefox Add-ons | Homepage
Formerly known as Read It Later, Pocket saves selected web pages for reading them later, just like a bookmarking system. You will have to create an account for Pocket. For offline saving you have to check the Automatically make all saved pages available offline option in Preferences.
SearchPreview | Install from Firefox Add-ons | Homepage
This add-on inserts thumnail previews for the Google, Yahoo and Bing search engines, as well as popularity ranks.
Flashblock | Install from Firefox Add-ons | Homepage
As the name suggests, Flashblock will block all Flash content from loading, and provides the web page with an empty placeholder in the place of the Flash content. It allows to display the Flash content with a single click. One of the most useful extensions in my opinion. It also comes with a so-called 'Whitelist', in which pages allowed to display Flash can be added. I couldn't get it to disable Flash after clicking Allow Flash from this site (except from the Preferences window).
Advanced Users/Web Developers Add-onsFirebug | Install from Firefox Add-ons | Homepage
NoScript | Install from Firefox Add-ons | Homepage
Greasemonkey | Install from Firefox Add-ons | Homepage
Greasemonkey is a very powerful extension which extends the functionality of Firefox by using various scripts which can be downloaded from http://userscripts.org/. This is a must-have extension if you want to extend Firefox and customize it the way you want.