SEO tools can help with analyzing websites for competitive advantage. The best SEO tools are those that remain within the white hat realm of ethical, Google-compliant practices.
The best place to start understanding SEO (search engine optimization) is at:
SEO at Google Webmaster Tools
Google-Friendly Sites at Google Webmaster Tools
Webmaster Guidelines at Google Webmaster Tools
Link Building (Link Schemes) at Google Webmaster Tools
Canonicalization (Setting a URL as the Preferred Version of a Content Page) at Google Webmaster Tools
Google provides authoritative, official information these and many other SEO topics.
Google Analytics can track activity on your site in a wide range of metrics and it's free.
Once you understand Google's viewpoint on a wide range of SEO issues, you're ready to investigate SEO tools and tool suites for managing SEO for your own website or those of clients.
You can also learn a lot from SEO podcasts, such as are found on WebMasterRadio.fm. My favorite shows there are SEO 101 and Webcology.
Here are some recommended SEO tools you can look into and decide if they're a good fit for what you're doing.
(1) Majestic SEO - Claims to have the largest database of how websites are linked to each other. Site Explorer, Keyword Checker, Link Profile Fight, Backlink HIstory, Bulk Backlink Checker, Clique Checker, Majestic Million, Comparator, Neighborhood Checker.
(2) Raven Tools - Suite of internet marketing tools. SEO, Social Media, PPC (pay per click), Metrics.
(3) ebrandz - Internet marketing tools. SEO, PPC, Social Media, Conversion Rate Optimization.
(4) SEObook has a set of free and paid SEO tools.
(5) SEO Power Suite offers Rank Tracker, Website Auditor, SEO Spyglass, and Link Assistant. Some SEO gurus advise practitioners to avoid Link Assistant as it has allegedly sparked accusations of URL harvesting and spamming. It's a tool that identifies potential link exchanges, but Google doesn't like too many reciprocal links.
(6) Optify provides digital marketing tools for leads management and more.
(7) Authority Labs - White label SEO tools that you can resell to your clients. Check out the Authority Labs blog.
(8) SEOmoz has a suite of tools that includes Campaign Manager, Followerwonk, Link Analysis, On Page Analysis, Rank Tracking, Crawl Test, SEO Toolbar, Keyword Analysis, Mozscape API.
Also see Search Engine Land "Enterprise SEO Tools: The Marketer's Guide", Schema.org for HTML code to help your website's structured data and databases communicate with search engine spiders, and W3C Validator to check your website's compliance with established web standards.
SEOmoz "Using White Label for SEO".
StepForth Web Marketing "Red Zee Top3 Promotions Review".
Marketing Land "Return of the Google Dance".
Search Engine Land "Periodic Table of SEO Ranking Factors".