IMO the only tools that are really "essential" for SEO are a brain and a spreadsheet but I'll rattle off a few others I find myself using on a regular basis.
For initial site audits i'll use a spider tool like
http://www.auditmypc.com/free-sitemap-generator.asp (FREE) or
http://www.screamingfrog.co.uk/seo-spider/ (£99 per year). Other folks swear by Xenu Link Sleuth but I prefer the other two. Dump the results from them to a spreadsheet and you'll get a good overview of the site architecture, onpage optimisation and quickly be able to spot SEF issues like dupe content, dodgy status codes like 404/500/302's etc
It's easy to go overboard with keyword research when you get into the paid tools. I find a solid brainstorm followed up by some metric validation from the popular free tools to be a good way to go. If it's a topic you're not familiar with then tools should be able to throw up some relevant jargon. Not a bad idea to do a bit of keyword frequency analysis of articles/pages/forums that exist on topical authorities, I tend to use
http://www.writewords.org.uk/phrase_count.asp a bit, if you're visual then maybe dump it into wordle.net to come up with a word cloud.
SEO for Firefox is great for SERPs analysis:
http://tools.seobook.com/firefox/seo-for-firefox.html
As far as free backlink analysis tools go since the demise of Yahoo Site Explorer,
http://www.backlinkwatch.com/ isn't too bad. If you're looking at a paid option, then Majestic SEO, ahrefs.com and OSE are quite powerful.
We had a great roundtable session last year about SEO tools at the Brisbane SEO Meetup, you can read the round-up here:
http://seomeetups.com.au/seo-brisbane/list-of-our-favourite-30-seo-tools/
I'll leave it at that for now. By no means a comprehensive guide, and individual tastes are bound to vary but I hope that gives you some ideas.