If you are looking for a free keyword popularity tool, the first cab off the rank by a long way is the Google keyword tool.
This research tool enables you to quickly get a snapshot of whatever you may have in mind to study.
When you bring the free key word tool up in your browser (just type in -Google Keyword Tool) type in your selected keywords then make sure that you check the box for synonyms, and when the search results are returned, go to the box titled "Choose columns to display" open up the options, and go to the bottom and select All.
This will now show the results for not only your selected keywords, but those keywords that Google believes are related.
Data retrieved will include the search volume, estimated Cost per Click and advertising competition, you can then use this to judge what best to optimise your website with, in order to gain the best search engine optimization results.
Should you know the web addresses of your immediate competition, then you can also use the Google keyword tool, but instead of entering keywords, instead check the box on the left to get website content.
Google will then provide a list of what it believes your competition is about.
This is also great to enter your own website URL, if Google returns with off target keywords, then you have a real good base from which to go to work and improve your own search engine optimization.
There are a huge number of other free key word tools, two I recommend being wordtracker and Traffic Travis.
The later provides a lot of data for a free service.
Finally as with all free tools there are limitations, plus the majority of people will be relying on the same data.
To get a jump on my opposition I use the free tools as a starting point only, then look to other paid keyword tools in order to diversify.