Essentially, PC works the following way:

• Visit a website that offers a link to download a simple computer program. 
• Download and install the computer program .
  
• This computer program will look and behave as if it were a screensaver.
• Data from the philanthropic computing organization will be sent over the Internet to your computer. 
• When the computer is not being used, this “screensaver” will run. 
• When this screensaver is running, your computer's resources will be used to compute the data that the organization has sent to you. 
• After your computer has finished computing its assigned data, the computer will send the computed data back to the organization via the Internet. 
• The process that has just been described is known as distributed computation. 

For further clarification, see the link below.
ExtremeTech  - This site provides a good explanation of philanthropic computing and it includes a diagram of the process.  It is the best I have seen on the web.

The following links describe how the featured projects work in detail.

SETI@home

MoneyBee

Lifesaver Screensaver (This site will not identify itself as Lifesaver Screensaver.  It will be listed as Grid.org)