A list of free SEO tools:

Keyword related:

Wordtracker keyword tool (free trial)
http://www.wordtracker.com/

Digital Point keyword suggestion tool - can search by country
http://www.digitalpoint.com/tools/suggestion/

SlowSEO keyword selector tool that shows cost per click information
http://www.slowseo.com/tools/keyword-selector-tool.php

Good Keywords downloadable keywords tool
http://www.goodkeywords.com

Link related:

Marketleap’s Link Popularity check, Search engine saturation and keyword verification tools
http://www.marketleap.com/publinkpop

Sitemap creation tool
http://www.xml-sitemaps.com

Google webmaster tools - create a Google account to access these tools - shows diagnostics, statistics, link report, keywords and submit your sitemap.
http://www.google.com/webmasters/tools

Google Toolbar –quickly and easily view page rank, keyword density and back links
http://toolbar.google.com

Assorted:

Submit express have a handy toolset including a page snooper
http://www.submitexpress.com/snooper

A load of seo tools including rank checkers for most search engines
http://www.searchenginegenie.com/seo-tools.htm

Bruce Clay has a free handy toolset including keyword activity, keyword density analyzer, competition research and link analysis. You can also get access to more tools by subscribing for free.
http://www.seotoolset.com/tools/free_tools.html

Google Duel - Good to compare two words, such as plural to see which will rank better.
http://www.googleduel.com/original.php

Screen Res comparison tool - More of a web development tool but extremely useful
http://www.markhorrell.com/tools/browser.html

SEOChat keyword difficulty check
They also have a number of other tools available
http://www.seochat.com/seo-tools/keyword-difficulty

W3C Markup Validation Service, a free service that checks Web documents in formats like HTML and XHTML for conformance to W3C Recommendations and other standards
http://validator.w3.org

Domain age checker- Allows you to check multiple domains at a time
http://www.webconfs.com/domain-age.php

Link Vendor have a nice tool that shows links on a website with information including linked page’s PR, whether it’s an external link and if the no follow tag has been used.
They also have a decent free tool set
http://www.linkvendor.com/seo-tools/visualpagerank.html

Webcomfs SEO friendly redirect check, they also have many more free tools:
http://www.webconfs.com/redirect-check.php

Seo Digger
http://seodigger.com

Firefox SEO plug ins:

These plugins display a toolbar with just about every thing you could wish for in one easy snap shot, including PR, backlinks, domain age, alexa rank, DMOZ inclusion, whois and much more.

SEOpen
http://seopen.com/firefox-extension/index.php

SEO for Firefox:
http://tools.seobook.com/firefox/seo-for-firefox.html

SEOQuake:
http://ff.seoquake.com/?sln=en

 

Video Distribution:

Tube Mogul
http://www.tubemogul.com

 

Analytics:

Click Tale
http://www.clicktale.com

Google Analytics
http://www.google.com/analytics/

 

Testing:

Google Website Optimiser
http://www.google.com/websiteoptimizer

 

Article Distribution:

Articlesender
http://www.articlesender.com

 

PPC Tools:

Adwords Wrapper
http://www.mikes-marketing-tools.com/cgi-bin/adwrapper.cgi