identifying Keywords 

We recommend the following tools to help you identify keywords. They are either free or pay with limited free component:
Google AdWords Keyword Planner
Bing Keyword Research
Ubersuggest
Keyword Discovery
Wordtracker
Google Trends
We have the following tool suggestions to discover keyword density:
Internet marketing ninjas
Dave Naylor
The Moz On-Page Grader
For the Enterprise, we recommend this additional tool:
Hitwise Consumer Intelligence

Google Toolbar

The Google toolbar is now available only for the Internet Explorer browser. And it's understood that Google PageRank, or at least the accurate display of it to users within the Google Toolbar, has not been updated for sometime. Nevertheless, for historical reasons, it's good to have the free Google Toolbar downloaded for the IE Browser.
Wikipedia has a good entry on the PageRank concept.

analysing competitor backlinks

We recommend the following tools which have both free as well as pay plans:
Some simple strategies you could consider with any of the above tools:
  • a. Enter your URL
  • b. Find out how many websites link to you. Also who they are.
  • c. Now test your competitor's site.
  • d. Find out who are all linking to that site.
  • e. Target them.

link and traffic building tools

We have heard good reports about the following tools for link building:

Broken Link Building from Citation Labs
Link Prospector, also from Citation Labs
StumbleUpon PaidDiscovery
Promoted Tweets from Twitter
Buzzstream for LinkBuilding

Sign up for google analytics

Sign up for Google Analytics, Google Webmaster Tools, and Bing Webmaster Tools. These are free tools, but pack great punch.
Onelook.com aggregates definitions from 1,000 dictionaries. You can discover lots of words using wildcard searches in the formats they've given. The reverse dictionary of Onelook is a useful place to search if you're lost for a word, but know the concept.

try an Alternative to Google Analytics

Besides Google Analytics, Clicky comes with good recommendations. Also you can use statcounter. An advantage is its provision for exit page analysis.

beyond tools, it's skills which matter

We have only briefly mentioned a few of the tools we've used and heard about. Search Engine Optimization is not accomplished by purchasing the most powerful tools which money can buy. It's a craft which requires years of training, the readiness to constantly learn and update yourself, and the humility to realize that you may not always have all the answers. Good SEO practitioners derive the most pleasure when they are able to work with their clients and solve their pressing problems. Beware of fakesters who come offering secret recipes to catapult your website to the top organic rankings. It's never that simple. You have to keep working at it.  Best of luck.

mega list of free and near-free tools