Using the good ol' free Google Keyword Planner is great. But there is an alternate universe of Long Tail keyword phrases that live in Google's other database--namely, their "Autocomplete database." That's the database that is accessed when you're typing queries into the Google search box (notice that list of suggested keywords that pop up as you type?)
The keywords you want to focus your SEO on should closely resemble the vocabulary of your audience. In order to come up with the proper keywords you really have to get inside the heads of the people who search for your website. What terms will people use? How do people search? Which question does your website answer? You should create a list of all search terms people could use and think of combinations and nuances within these search terms.
Google's AdWords Keyword Planner tool is another common starting point for SEO keyword research. It not only suggests keywords and provides estimated search volume, but also predicts the cost of running paid campaigns for these terms. To determine volume for a particular keyword, be sure to set the Match Type to [Exact] and look under Local Monthly Searches. Remember that these represent total searches. Depending on your ranking and click-through rate, the actual number of visitors you achieve for these keywords will usually be much lower.
TIP: A really good strategy for increasing your search engine rankings (and maybe even getting a featured snippet), is to pick a number of popular questions, and answer them in your content. You can do this in the form of a ‘Question & Answer’ section or maybe ‘FAQs’. Just pick half a dozen or so questions, and list them, together with a short answer.
Search Analytics can be found under the 'Search Traffic' section and provides details of the keywords that drove clicks to your website, based on data for up to the last 90 days. The difference with Google's version is that you can filter the data to put extra context around the keywords, such as filtering by country to see keyword popularity based on country, which can be useful when carrying out keyword research for websites that service more than one country.
Long Tail Pro is a keyword research tool that helps you quickly find low competition keywords for your website. In particular, the tool is built for people doing SEO (search engine optimization) as opposed to people doing paid ads (Google Adwords PPC). I'm sure there are some advertisers that use the tool for paid ad keyword research, but the tool was built for SEO keyword research.