Neat bitcoin technology application: "wisdom of the crowds" decentralized prediction market. Apparently when you average people's guesses about an outcome, and they have a stake in making that prediction right, the average is usually pretty good, whether it be for jelly beans in a jar or a lost submarine: http://wisdomofcrowds.blogspot.com/2009/12/introduction-part-v.html
I'm still a little foggy on why the decentralized part matters,... I guess it comes down to trust. For the same reason you don't want a government or bank holding the ledger in Bitcoin, you probably don't want a single entity controlling a prediction result or compensation--too much incentive to tweak the results for personal gain.