Google AdWords Keyword tool
The first step in any SEO campaign is identifying the keywords/key phrases you want to use - popularly known as keyword research. Google AdWords Keyword tool is the most widely used and acclaimed free tool for keyword research. It will let you know which keywords to target, the competition for those keywords, and will also offer suggestions on popular phases you might not have even thought of. There are other paid keyword research tools like keyword discovery and wordtracker which are good as well, but this tool is valuable and efficient when it comes to a free SEO tool.
XML Sitemap Generator
If you place a formatted XML file with site map on the web server, you enable Search Engine crawlers (Google) to find out what pages are present on your website and which ones have been changed recently. Whenever you add new content to your site you will obviously want to make sure Google knows about it. Sitemaps help search engine's spiders crawl your site and understand what it's about. The XML sitemap generator will create a sitemap compliant with the search engine that you need to upload to your home directory every time your site is updated.
SEO Book's Rank Checker
A Firefox plugin that enables you to check where your website is ranking in keyword searches. You can save details of every campaign and run reports from your browser in seconds. It helps you analyze website rankings on the three major search engines - Google, Yahoo and Bing. SEO Book's Rank Checker is a widely used tool which is highly recommended to check your search engine rankings.
Hubspot's Website Grader
This free SEO tool can perform a health check on your website before awarding it a grade and offering advice on how it could be improved. A few interesting features are blog analysis, indexed pages analysis, metadata, domain info, inbound links, last crawl date, social media grader etc. You can discover how effectively your site is described (language) and how often it has been bookmarked on Digg or Delicious. Also, you can add a Website Grader widget (which provides you with a code to paste into your site) displaying a badge featuring your search marketing quality score.
SEO revolves around backlinking. This tool gives you a clear count and analysis of your website's backlinks including the page they are on, anchor text and the total number of backlinks on that particular page. This useful tool will let you know who's linking to your content and track the success of your link building campaign.
Google Analytics / Google Webmaster Tool
These two tools together make the fundamentals of search engine marketing. Based on Urchin, Google Analytics is a top analytics tool available to everybody for free. This analytics gives you details about traffic behavior, funnel information, conversions, landing pages and much, much more. Google webmaster on the other hand gives you a view of what Google "sees" on your site including crawling speed, crawling rate, highest pagerank of your site and more.
SpyFu
SpyFu lets you enter either a keyword or domain into its search box to begin the process. You are afterward presented with results to help you decide if a particular keyword is an appropriate choice for your SEO or ad campaigns and other keyword possibilities. From domain searches, you can see your competitors and discover details of their organic or paid search campaigns. There is paid version of spyfu.com which offers more features, but the free one has quite a lot as well.
Compete
This is a great tool for analyzing your competition. You can see traffic and engagement metrics for any website; identify sites for affiliate and SEO link building. Also you can analyze competitor keywords and search strategies for free. Other features include; analyzing subdomains, tagging, comparisons, portfolios and CSV download of information. Compete.com also has a subscription based paid service which you can use to attain additional features.