Chrome

I suggest using this for devs or advanced users only. Ones that won't freak out completely if their browser crashes and they loose some data. This is solely because Text Area Cache is way better than Lazarus on Chrome. 

See the full list by searching the chrome web store. 

Users

• Lazarus
• Single Click for Google Apps (yeah, not a Salesforce extension but gives you back your missing black bar functionality in Gmail). 

Admins

• Salesforce.com Sandbox Favicon - So useful. Firstly it adds an S on the favicon when you are logged into a Sandbox - so helpful! Secondly it adds the instance of Salesforce to the titlebar (eg if you have a custom URL it will show (customurl) in front of the regular title. 
• Salesforce.com id Clipper
• Salesforce.com Enhanced Formula Editor - I find this one both totally annoying and totally useful. I can't decide whether to have it turned on all the time or not. Whenever I have it turned off I miss it, when I have it turned on it annoys me. New features mean that it is less annoying though, so worth having.  
• Quick Login As - log in as other users in your Salesforce Org.
• Grey Tab - OMG, how cool is this app - never be without it. I always add "system" type fields to the page layout and clutter up the page layout. This tool just shows every field.  
  • It even shows the full HTML of rich text fields!
• Salesforce Inspector - just about as OMG as Grey Tab! See Salesforce Inspector for notes. 
• Salesforce Admin Check All! OMG how useful is this Extension - Check ALL boxes in the Enhanced Profile Editor! 

Advance Admins or Devs

• Salesforce Navigator
  • See the excellent post Force.com Development Tips & Other Snippets by Matt Lacey: Flying Like A Pro: Salesforce Navigator for how to use Navigator. 
• Web Developer
• LastPass or 1Password - a Must for logging into multiple orgs!
• SimpleExtManager - turns extensions on and off - very useful for when you don't use Salesforce all of the time. 
• Wappalyzer - Not Salesforce related, but useful for devs - look at any website and it tells you at a glance what it is made of. 

Other Ideas

Also see the thousand blog posts on different user's ideas of what extensions to use. 

• Page:
  Force.com Development Tips & Other Snippets by Matt Lacey: Flying Like A Pro: Salesforce Navigator
• Page:
  Useful Google Chrome Extensions for Salesforce
• Page:
  » Chrome Extensions to Cure your Pet Peeves
• Page:
  6 Great Salesforce Chrome Extensions - Cirrus Insight
• Page:
  5 Extensions to Optimize Google Chrome for Salesforce Admins and Developers - The Cloud to Cloud Backup Blog
• Page:
  Top Google chrome extensions for Salesforce
• Page:
  Chrome Extensions for Salesforce Administrators
• Page:
  Chrome Extensions Every Salesforce Developer Should Have
• Page:
  Top 5 Salesforce Chrome Extensions - Salesforce Ben

Firefox

I would use Chrome for admins or devs and use Firefox only for Users. 

Users

• Australian Dictionary https://addons.mozilla.org/en-US/firefox/addon/english-australian-dictionary/ 
• Text Area Cache https://addons.mozilla.org/en-US/firefox/addon/textarea-cache/ 
• Snap Links
• Highlighter https://addons.mozilla.org/en-US/firefox/addon/highlightall/?src=search