Here's the way the NWS defines it:
Forecasts issued by the National Weather Service routinely include a "PoP" (probability of precipitation) statement, which is often expressed as the "chance of rain" or "chance of precipitation".http://www.srh.noaa.gov/ffc/?n=pop
ZONE FORECASTS FOR NORTH AND CENTRAL GEORGIA
NATIONAL WEATHER SERVICE PEACHTREE CITY GA
119 PM EDT THU MAY 8 2008
INCLUDING THE CITIES OF...ATLANTA...CONYERS...DECATUR...
119 PM EDT THU MAY x 2008
.THIS AFTERNOON...MOSTLY CLOUDY WITH A 40 PERCENT CHANCE OF
SHOWERS AND THUNDERSTORMS. WINDY. HIGHS IN THE LOWER 80S. NEAR
STEADY TEMPERATURE IN THE LOWER 80S. SOUTH WINDS 15 TO 25 MPH.
.TONIGHT...MOSTLY CLOUDY WITH A CHANCE OF SHOWERS AND
THUNDERSTORMS IN THE EVENING...THEN A SLIGHT CHANCE OF SHOWERS
AND THUNDERSTORMS AFTER MIDNIGHT. LOWS IN THE MID 60S. SOUTHWEST
WINDS 5 TO 15 MPH. CHANCE OF RAIN 40 PERCENT.
What does this "40 percent" mean? ...will it rain 40 percent of of the time? ...will it rain over 40 percent of the area?
The "Probability of Precipitation" (PoP) describes the chance of precipitation occurring at any point you select in the area.
How do forecasters arrive at this value?
Mathematically, PoP is defined as follows:
PoP = C x A where "C" = the confidence that precipitation will occur somewhere in the forecast area, and where "A" = the percent of the area that will receive measureable precipitation, if it occurs at all.
So... in the case of the forecast above, if the forecaster knows precipitation is sure to occur ( confidence is 100% ), he/she is expressing how much of the area will receive measurable rain. ( PoP = "C" x "A" or "1" times ".4" which equals .4 or 40%.)
But, most of the time, the forecaster is expressing a combination of degree of confidence and areal coverage. If the forecaster is only 50% sure that precipitation will occur, and expects that, if it does occur, it will produce measurable rain over about 80 percent of the area, the PoP (chance of rain) is 40%. ( PoP = .5 x .8 which equals .4 or 40%. )
In either event, the correct way to interpret the forecast is: there is a 40 percent chance that rain will occur at any given point in the area.