The princip
If you own Windows XP or Vista, try this (no risk) : make a right-click on the taskbar and choose « Taskmanagement ». In « Performances », look at « Use of the UC » or « Use of processor ». I think that the indicate number stays under 10 %. In fact, if you're not watching a movie, playing or executing a complex program, this number is almost always under 10 %.
What does it mean ? It's the power of your computer that you are using : 90 % of the power of your processor isn't used. Multiply this by the millions of computers in the world and by the hours of use and you obtain an enormous power of calculation, a lot bigger than the one of the best super-calculators of the NASA. There is on Earth a great deposit of informatic ressources and it's almost not used. The programs of distributed computing have the vocation to benefit from this.
The princip is simple. for example, a team tries to modelize the influence of temperature on the repartition of the clouds. But a lot a parameters (density of the clouds, parameters that have an effect of the thermal conduction, ...) are not very well known and the team has to make the calculations with many values of these parameters, sometimes millions of. So there are billions of calculations to do and the super-calculatros are rare : nothing says that the team will can use one (there can be more interesting projects) and even if it's the case, the price can be too high... The problem can be cut in small independant parts (for each value of the parameters) and the idea of the distributed computing is to offer volunteers to make their computers available to a part of the calculation. This method can even be faster than the other one.
If you want to try, you should first sign-up in a website and download and install a small program, which is common for all users. Then this program downloads from the server a serial of parameters (at this moment, you need a connection to Internet), it makes the calculations alone (at this time, you don't need a connection anymore) and, generally after a few days, it sends the results to the server. For you, there is not any disadvantage : the datas are in general little and the connections of the program only take a few seconds, so they don't use all your time on Internet. Moreover, the program works in silence : it starts and stops with the computer and will appear only when it will want to send or receive some datas.
For the use of your processor, it's discreet too : the program doesn't use all the available power (because it could stop the computer) and if you launch over applications for your personal use, it gives you the priority. You also can snooze it when you need to. So, this kind of programs doesn't bother (I tried myself).

The programs
If you want to offer a little part of your power, here are different programs (this list is not exhaustive) :
- World Community Grid : in fact, this project groups many others together. A same program serves different researches but you can choose to which you participate. Address :
- SETI@Home : it's probably the more famous because it's one of the oldest. The project consists in studying some recordings made in Arecibo (the telescop that has been made famous by X-Files or movies like Contact) and filtering the datas in hope to find the manifestation of an extraterrestrial intelligence. This project is slightly different from the others : the others are finite, there are a known quantity of calculations to make and the project will one day be terminate, whereas SETI@Home doesn't have any end. Address :
- GIMPS : It's a program to find some prime numbers. Address :