There are many different types of SEO tools performing different functions. There are also platforms which perform nearly every task you could think an SEO would need to do. I will attempt to go over most of the tools used in SEO by professionals. I will skip some of the larger agency platforms because of the costs.
Lets talk briefly about what different tasks SEO tools perform.
Keyword research is about figuring out what terms people are using to find the service or information you are providing. The tools for keyword research have various functions as well.
There are many tools which function only to find the largest amount of keywords on a particular topic availabe. These generally find related terms by using Google or another search engines related terms. Examples of these tools are ubersuggest, keywordshitter, & seoquake. The original would of course be the Google adwords keyword planner. Bing also has a very nice version of the keyword tool available.
Finding Search Volumes
Although there are many tools which will show you search volume now one of my favorites is the keywords everywhere browser plugin. Tools like Ahrefs and SEMrush also have there own ways of determining search volume. One of my favorite things about the keywords everywhere plugin is the price. The Google adwords keyword tool always gave you the search volumes for free until mid 2017, and it is awesome that a tool came up to fill the void so quickly when they stopped.
Finding Keyword Competition
Keyword competition is very tricky to calculate. We can look at metrics from various tools all day long and still not know why Google decided to put a certain site in the #1 position. It's always a measure of relative guesswork to find terms which are good targets for ranking. In the past I used the Moz browser extension which gave a value for competition when you performed that search in Google. Once they stopped allowing people to see that value for free that toolbar was removed from my browser.
The tools I use the most to check the keyword competition scores are kwfinder, VidIQ for YouTube. I like to look at the serps myself as well as look at analysis from semrush and ahrefs to get a good idea of competition.
There are plenty of tools that can do SEO audits of your website. They will crawl your site and let you know what errors you have. Ways you can further clean up your site techincally and things you are missing.
My favorite for this is simply Screaming Frog SEO Spider. It does a crawl and allows you to look at individual issues you may not have known about. It will point out 404 pages, missing meta descriptions and much more. There are others which are similar but Screaming Frog is the original.
If you just want to look very quickly and you're not on your own machine tools like seocentro, or seositecheckup can perform quick audits of your website as well.
While tools to build links isn't generally advised going directly to your main websites there are tools which can help you find link prospects as well and automate outreach for links.
Link Outreach Tools
The industry standard for link outreach has got to be pitchbox. There are some compeditors coming into the market as well now. One that is getting attention from some big names is Ninja Outreach. I did get a chance to use it and found it worked pretty well. There are other places which have lists of blogs that accept guest posts as well. I have a membership in a few of these. I gave it a shot and although I thought it may be a bit saturated with spammy type websites that accept almost anything it is anything but that.
Automated Link Building
Automated link building tools are things like GSA search engine ranker, SEnuke, RankerX, and many more. These post links on various sites according to your settings on each project.
These are commonly talked about as spam tools but they can be use to create higher quality links as well.
Web 2.0 Posters
RankerX, GSA, and SEnuke can be used as web 2.0 posters but they also have some additional features. The tools I'm talking about here are things like Syndwire, Source Machine, FCS Networker, Rankwyz and Money Robot.
These tools post to free web 2.0 sites that you set up in them and some of these tools even create web 2.0 accounts.
What do I mean by link indexes? Well these are things like Majestic, Moz, and ahrefs which keep track of what websites have links pointing where. There are many more such as SEMrush and Linkprofiler but the largest indexes are from the first three mentioned.
These give us values of the strength of different websites based on their link profiles. This can be useful to see how many links a compeditor has, or how many links are pointed to the websites showing up on the first page of a particular search.
Links are still one of the top ranking factors, if not the most important certanly in the top 2. Having links isn't enough alone. It's important to have links from websites that are authorities. How can you tell a site is an authority? Well the sites that have links from other sites with lots of links are authoritites.
Websites like IBM or Forbes have tons of links from smaller websites as well as other large websites. The link index tools give us things like Domain Authority or Trust Flow which is that particular tools measurement of strength of the incoming links.
Link indexers are a bit less notable but still very useful. Most links will get indexed on their own but when links are build on web 2.0 sites or with other automated tools it can be harder to get those indexed. Google's index is huge and while they want to have as much information available to people as possible they also want to show people quality results in the serps.
This means if websites don't have many higher quality links they may not be noticed by Google. So indexers use these high quality sites to ping links to Google basically. This is at least one way that they have worked in the past although there are probably newer versions using different techniques now.
Free Tool Collections
There are a few websites which have collections of tools built that you can download and run on your machine for free. My two favorite sites like this are supagrowth, and ranktank.
This website has a large collection of tools for scraping expired web 2.0's, cleaning email lists, finding hidden pbn sites, and much more. This collection is by far my favorite.
This website has a collection of spreadsheets which use API keys to see different aspects of SEO. One of my favorite of these tools is the rank tracker as it's free and pretty simple to paste a list of keywords into a spreadsheet for checking.