Many posts on this forum mention tools but some are self promoting. I just want to compile a list of sites where one can use to obtain SEO/competitive intelligence insights? (Better if there are free ones out there).
For starters, I like using Google's AdWords and Google's Trends.
What do you use?

First you let us know which kind of analytic you want to do, like related to keywords, on page, traffic, and general coding

i am mentioning here soe tools which i used :

https://adwords.google.com/select/KeywordToolExternal (for keywords)

www.google.com/websiteoptimizer (for coding and design analysis)

www.websiteoptimization.com/services/analyze/ (for coding and design analysis)

www.google.com/analytics/ (for traffic)

http://www.seoworkers.com/tools/analyzer.html (for seo analysis)

Both SEO Elite and Keyword Elite from Brad Callen are very good.

piwik.org is a open source PHP based web analytics tool. If you want your data to be NOT hosted with external host then it can be really worth ...

Read more about Piwik

Hmmmmmm This sounds good..

Wow, that is a good list. Thanks. I will also add using quantcast to get a glimpse of behavioral demographics and keywords that drive traffic to other sites.

To analyze my competition, I rely on SEO Elite. For keyword research, I use the free Google keyword suggestion tool. When it comes to analyzing stats, Google Analytics is the best tool out there.

Its really critical that you do SEO Analytics correctly so you get google juice & more traffic...

I recommend that you buy Market Samurai or Keyword Elite to do SEO Analytics, this tools would really help with complete this task easily & effectively...

I am using Google analytics, WebCEO, Compete, Quancast and different Firefox Addons.

I have used piwik.org and it is little bit confusing to me.