There are a number of free tools available online that can help with keyword analysis. Here are just a few of the most commonly used:

Google AdWords Keyword Tool
This remains a highly popular tool for generating keyword ideas related to your product, service or topic.

Wordtracker Free Keywords
This does the same thing as AdWords but with less option to refine your search.

Keyword Research Tool
Found on Seobook.com and powered by Wordtracker, this tool provides search figures across the major search engines, Google, Yahoo and MSN and includes click-throughs to several keyword suggestion tools.

Quintura
The Quintura tool will give you an insight into keyword maps based on popular search terms. Can be useful in brainstorming keyword ideas.

Blogpulse
Blogpulse lets you track trends on keywords, phrases and links by monitoring the buzz surrounding these search terms. Measured in percentage of blogs posted on those subjects within your specified time frame.

Longtail Keywords Tool
Find out what tail terms are suggested for your site or ads using the Longtail Keywords Tool. Can be a bit picky and does not always yield results but can give you an idea of what your competitors might target even if your site doesn’t show up.

Digital Point’s Keyword Tool
Gives keyword suggestions for your topic with options to overlay specific regions for localised results.