BOINC is a platform for distributed processing. For example, it takes a lot of processing time to sift through radio signals from the whole sky looking for possible alien signals, like it takes a lot of processing time to test various iterations of mathematical models to see what fits physics data the best. You can volunteer your PC to work for these projects when it is idle, and your PC is idle really, most of the time.
I volunteer for the following projects:
Einstein@Home - search for pulsars
LHC@Home - help develop particle accelerator physics
SETI@Home - search for alien radio signals
Cosmology@Home - match math models to physics data
As you may know, I have an awesome PC with 8 virtual cores and 6Gb RAM :_)
Here's a picture of BOINC using it hard.
You can see most of the processors are close to max (it often does max them all) and RAM usage dip suddenly as I logged on to actually use the machine.
Here's the BOIC control screen:
You can see the actual project tasks that are running, ready to run, finished and waiting for RAM allocation.
This is all very cool, but it also keeps my office toasty warm in the winter.
[update - Rosetta@Home also seems very worthy - thanks Rich]