The disavow tool simply allows you to tell Google to ignore certain links to your site. Lots of poor, spammy links can negatively affect your search ranking, and this tool, in theory, allows you to remove the effects of those links.
At least one reason it was introduced (according to many) was to combat the practice of negative SEO. If you want to harm a competitor, you can go get 10,000 spammy links to their site for $19.95 from a bulk SEO service, and basically penalize them out of the search results. There was a long time discussion about whether this was possible. Many SEOs see the disavow tool as essentially an admission that such negative SEO practices might work, and have thus given you a tool to combat them.
If you are considering using the disavow tool, you should tread cautiously. There's a great review of the tool on SEOMoz that goes over how it works, what the potential ramifications are, and who should consider using it: