Distributed computing involves using software to network thousands of individual computers together into, effectively, one huge hive computer. In this case, the BBC is asking you to donate your spare computer cycles to help predict how the climate is going to change. Climate prediction requires huge, huge, huge amounts of resources - but helping out here won't cost you a thing.
The distributed computing software kicks in when you're not using the computer, and only then. So this is a form of recycling - your waste is going to huge benefits for everyone. There's absolutely no reason not to do it - so, go to! [GT]
Well done, by using your 'spare' cycles, your PC will not go into standby (or power saving mode) therefore using more power and directly contribute to climate change.