A. Desktop Tools
Spiders your site and helps identify broken links and redirects.
Like Xenu, Screaming Frog spiders your site for SEO elements and errors.
Yet another spider.
B. Online Tools
A classic tool, though rumor has it it wont be around much longer. Helps review/analyze list of cashed pages and inbound links.
Helps determine the popularity of a link, and does backlink analysis.
Analyzes your link profile.
Allows you to inspect the HTTP/HTTPS headers of a page. Useful to seeing server responses and what not.
Helps identify duplicate content and establish uniqueness of text.
Provides some domain related statistics.
Gives you a preview of how search engine spiders see your site.
Tells you WHOIS information such as domain age
Gives you a plethora of data about your site, including inlinks and more.
Shows similar stats to Google Webmaster Tools (crawling, links, etc)
C. Firefox Add-Ons
Allows you to emulate different user agents (search engine spiders, mobile devices, etc) in order to discover stuff like cloaking.
Provides an SEO snapshot of the page via the SEO XRay Tool, quick site/page information in Lookup, and more.
Allows you to inspect and edit HTML code. You need this.
Helps review websites with IP based content serving (multi-language sites, etc)
Allows you to capture and analyze server header responses. Very useful for both analytics and SEO.
Lets you analyze the size of on page elements (images, CSS code, text/html, etc)
Lets you preview what Google search results are like in different geographical locations.
Shows a bunch of website related stats.