SETI: Search for Extra-Terrestrial Intelligence (SeTi here on wiki)
See: http://setiathome.ssl.berkeley.edu/
Not to be confused with the more mundane, but assuredly more silly STI project (http://totl.net/STI/) -- JohnBullock?
This project is the most impressive DistributedComputing initiative I have seen. ComputingPrimeNumber?s (PrimeNet?) is cool too, but they might have already done that. -- ChrisGarrod
Also take a look at http://www.popularpower.com/ Same idea as SetiAtHome, but the StickinessFactor is that they will offer to pay for your compute time. SetiAtHome only gives you your free alien.
Popular Power has apparently gone out of business, but appears to still be running their distributed network for non-profit customers. Other companies that appear to have disappeared in the pay for distributed processing space include Process Tree, Distributed Science, and Envive (which appears to have been acquired by Keynote).
Distributed Processing companies, generally operating on other models, continue to appear. Current entrants include United Devices (http://www.ud.com), Parabon (http://www.parabon.com), Entropia (http://www.entropia.com), Avaki (http://www.appliedmeta.com), Data Synapse (http://www.datasynapse.com), and (the newest entrant and still in stealth mode) Centrata (http://www.centrata.com). Most seem to be combining altruistic appeals (United Devices and Parabon both support cancer research) and prizes.
The latest standalone Seti@Home clone appears to be Folding@Home (http://www.stanford.edu/group/pandegroup/Cosm), which is experimenting with highly distributed protein folding. -- DavisFoulger
A problem with seti@home is that the program you download is not OpenSource. Therefore, you don't really know what your computer is calculating there. It could be simulations of explosions of atomic bombs. (Of course, if it was open source you had to read the code first and then check if it does what it should do and make sure that it can't do anything else but that depending on all possible input data, but the risk would be smaller nevertheless.)
Now, seti@home argues that this is for obvious security reasons. But obviously, if someone wants to hack seti@home, he can also do so when it is not OpenSource. It has been done before.
SETI@Home IS opensource nowadays. This wiki page is impressively out of date.
See also: TupleSpace, GenerativeCommunication, DistributedDotNet?