S3 is the technical term for a power saving mode that is more commonly known as "Suspend to RAM" (STR), or "Standby" in Windows. In S3 mode, power is only supplied to the system memory (RAM), all other components of the computer shut down. A computer in S3 mode often shows a blinking power indicator on the case.
The main benefit of this is that the computer can be put in standby and started up again in a few seconds, because all running programs and open documents stay open in the system memory.
The main disadvantage is that not all hardware and software fully support this feature. You may see applications freezing after starting up, or hardware not restarting. Another disadvantage is that if the computer loses power or is unplugged by someone who didn't notice the computer was not shut down, all information in memory (like unsaved documents) is lost.
Note: The S3 Standby Done Right guide has a gross miscalculation on the power consumption. Cameron calculates 4 kW as the idle power consumption. A good htpc uses about 40 W in idle, an older PC with bad power management is not above 100 - 150 W. So devide the result by 25 at least, better 100.