Free SEO Tools & Services

While tools like SemRush, Lipperhey, Moz and Raven Tools, etc. can be extremely useful for large scale SEO work, they all come at a price that not everyone can afford. They usually start around $60 or so, and often higher. Luckily there are a lot of free tools and services available, some of which offer excellent functionality. For administrators with only one or a few sites, these can be extremely helpful. The list below is in no way complete, and I will add new ones if I find any.

Free versions of paid tools

First of all, some of the commercial tools offer free versions with limited functionality, which still makes them pretty usefull. These are not limited time trial versions, so you can use them as long as you want. Usually they allow you to monitor one or two websites, but by removing checked sites and adding another one you can often use them pretty efficient:

In this category, checkout: Lipperhy (recommended!), WebCEO, SemRush, Majestic SEO and SEO Powersuite. All of these are web services, except SEO Powersuite, which is a desktop tool.

Free SEO webservices

There are hundreds of sites that offer SEO related services for free. Usually they are specialists, good at one area particularly, so for full SEO work you may need to use multiple tools.

• SEORCH.eu is a site offering multiple possibilities. It's main focus is on-page SEO checking. You can check one page at a time, but also checking multiple pages at once is possible. Other services are Google rank checking keyword suggestions and URL comparisons.
• Hubspot's Marketing Grader: This takes a slightly different approach, focussing mainly on online marketing issues, like social media popularity, backlinks, active blog-content, existence of landing pages, etc., which could be nice to have apart from full SEO audits provided by other tools.
• SEO Workers has an online SEO analyzer as well, though limited compared to SEORCH.
• Upcity allows you to generate a one-time audit of your site, provided you leave contact information (US only).
• Seoptimer.com: Limited on-line audits.
• InternetMarketingNinjas: A nice tool to check for broken links. It does an extensive crawl of your site, also checking redirects etc.
• Serps.com: Check rankings for any keyword. Also localized rankings per country.
• Backlink Watch: This tool looks basic, but detects your backlinks surprisingly well!
• My Site Auditor: A white-labelled SEO auditing tool, which you can offer on your own site (white-labelled)
• UpCity.com: You need to submit your details, and they will send you an SEO report
• Webmaster structured data testing tool: Test if your Microdata are configured correctly
• Webmaster mobile friendly test: Crucial to test if Google is able to correctly crawl your mobile / responsive site
• Zadroweb SEO Auditor: A nice SEO audit tool that also tests on some items other services don't test for. Still basic though.
• Quicksprout: An audit tool developed by Neil Patel, who was also involved in platforms like Kissmetrics (advanced Analytics) and Crazy Egg (heatmaps). The SEO audit does a really nice job.
• Siteliner.com: A tool from the developers of Copyscape to detect duplicate content issues and broken links on your site.
• SEO Analyzer: A tool from Neil Patel, a very popular blogger about SEO related topics.
• Brokenlinkcheck.com: Finds broken links in your website.
• ATOM SEO Broken Link Checker: similar to the previous line.

Free SEO Browser extensions

• SEO Search Simulator: Provided by SEO company Nightwatch. It provides a very easy way to chack anonymous rankings in Google, for different countries. You can store searches.
• Broken Link Checker by Atom SEO: This Chrome extension checks the dead links in the current page. 
• SEO Doctor: This is my favorite plugin (Firefox). It is always sitting somewhere, and I subconsciously keep an eye on it all the time while browsing. It helped me discover indexability issues on sites I developed a few times already (it shows a stop signal then). Also it gives you instant SEO scores on a scale from 1 to 100. It is a rough check, but often this is all you need.
• Moz Toolbar: This Firefox and Chrome extension is created by the Moz.com people. It is freely available, with limited features. Moz subscribers get more information. The free version already provides many nice features, like on-page SEO metrics, link popularity, social media counts, keyword difficulty, etc. It also provides an overlay in the Google SERP, showing additional information, like Page Authority and Domain Authority. Very nice is the possibility to set up different profiles for non-personalized Google searches, either global or country-specific.
• SEOQuake: This is a tool provided by SEMrush. Lots of information, showing you Pagerank, nr of pages for the domain in the Google Index, nr. of tweets, like and G+'s, SemRush value and popularity, keyword density warnings. Also it offers an additional option to get an on-page SEO report ith just a mouse click.
• SEO Status PageRank/Alexa Toolbar: Shows Google and Alexa Pagerank, backlinks and value.
• SEO Toolbar: A plugin developed by seobook.com, which has more plugins, amongst which some commercial ones. This one is available after registration, and it offers similar functionality as SEOQuake.
• RankChecker: This is also a plugin by seobook.com. It is a very nice plugin to keep track of Google rankings for stored sets of keywords, using non-personalized queries. 
• SEO SERP: Offers the same rank checking functionality as Rank Checker, but it is not for Firefox, but for Chrome.
• Open SEO Stats: Chrome only. Very usefull information regarding on-page SEO, Alexarank, Pagerank, Pagespeed, etc.

Free SEO Desktop tools

• Screaming Frog SEO Spider: This tool is incredibly handy for checking full sites for on-page SEO factors. It does not really grade them, it just lists them. It shows titles, metadescriptions, h1 presence, etc, for all URLs on the site you request. It allows for very easy checking for especially items like titles and metadescriptions, which often need human checking for relevancy. The tool is very quick to generate a nice view for this, and it can also export all data. Many SEO professionals use this free tool next to their commercial tools, as it adds a lot of value (see this survey, where it was chosen as the best all-in-one SEO tool). You only need a license if you need to check more then 500 URLs per site.
• Xenu Link Sleuth: This tools detects broken internal links and images, while it can also check external links.

Not directly SEO related tools, but still useful

• Spy on Web: Detect sites sharing the same IP / server
• Copyscape: Detect who is tealing your content
• Urivalet: Check HTTP-headers

And if you need even more, check here or here.