Bing Webmaster Tools
Similar to Google Web Master Tools. Think of this as the way you introduce your site to Bing. Bing will tell you errors it found when it crawled your site. You can also submit an XML Sitemap.
Google Analytics
This is a great tool. They also have Youtube tutorial videos on SEO. This is one of those tools that is probably more powerful than you realize. You can see demographical information on your visitors, visits, the pages people are visiting, where people are coming from, searches people entered to find your site, the pages they are leaving on, and way more. You can also merge this with Google Webmaster Tools so you see that information right here. You can also create great dashboards for clients or product owners. Piwik is a similar open source product.
Google Keyword Planner
This is a function of Google Ad Words. Or at least you need a Google Ad Words account to access this tool. But you can get great reports like this one. You type in a keyword or phrase and Google tells you similar phrases and keywords, how high the competition is for each string, and how many monthly clicks that string gets. I highly recommend you spend time with an SEO contractor analyzing keywords and strings to find a handful you want to rank for. There are some CMS plugins that assist with this.
Google Page Speed Insights
A great tool where you enter your URL into a field and Google tells you what you can do to improve your load times. All the suggestions are things you can do on your site, it doesn’t make suggestions like using a VPS or CDN.
Google Sitemaps Generator for WordPress
This is a plugin for WordPress that creates an XML file to submit to Google.
Google Web Master Tools
This is where you can submit your XML Sitemap and see errors the Google spider is finding. I highly suggest you fix these errors if you want to start improving Google results.
Keyword Suggestions
You enter a word or phrase and this gives you suggestions. It says it gives you 750+ suggestions, but I think you will quickly see some that might work and some you can skip. Finding alternative phrases that ranks high with searches but isn’t as competitive as the obvious phrase is a great strategy. Highly competitive phrases may be near impossible to rank for without a full time team, but there may be some less than obvious phrases you can try and get high traffic volume.
Moz Local Listing Score
Like it sounds, it gives you some indication how you rank locally.
Moz Open Site Explorer
This is partially free. You can see how high Moz ranks you, how many sites link to a site, and how many total links a site has. You can get up to 10,000 links if you subscribe to Moz, which runs about $100/month.
Piwik
Piwik is an open souce project that creates a website that acts like Google Analytics but just ofr you and your team. It is increadibly flexible, customizable, and hackable, but the learning curve is a little higher than Google Analytics. Some people like Piwik because they like the idea of keeping their traffic data secret from Google. Probably a good thing depending on your size. Unless you compete with Google, Google Analytics is proabably an easier solution.
Yoast
Great WordPress plugin for adding some SEO tools. One of the most widely downloaded plugins for WordPress. Great for helping content writers doing the basics to ensure content is optimized for search engines. Also generatex XML sitemaps. This will also preview how search engines list each article or page and helps you to edit to make sure it looks how you want.
YSlow
This is a Firefox plugin that explains what is slow and things you can do to improve load times. All of which is cool, but the Google Page Speed Insights probably replaces a lot of this.