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Is this something you can do on your computer?
Absolutely! That's the whole point of these kinds of things. The technical name is "shared computational power" or "shared computing." By using thousands of individual computers, sometimes more can be accomplished than by using one huge, very expensive, dedicated computer. And it's almost free for the researchers as we volunteers do this for nothing but the pleasure of helping science/health research. And you are correct SETI is uses the same principles.
Come on in, the water's fine! You don't have to be online 24/7. Don't even have to have the computer on 24/7. You'll simply get work done quicker if you do that and may see your 'rankings ' climb faster.