As I understand it, the Small Arms & Ammunition Manufacturers Institute (SAAMI) has established certain pressure limits for different calibers to be loaded to. There have been limits set above the standard pressures for certain calibers (.38, 9mm and .45). These are "+P" pressures. There are additional pressures levels above the +P for certain calibers (.38 and 9mm). These are "+P+" ratings.
The advantage of a "+P" or "+P+" round is that it generates a greater amount of velocity. Higher velocity tends to equate to better terminal ballistics (effect on the target) though several standard pressure loads deliver adequate terminal performance as well. In short, +P or +P+ ammunition generally equates to more "stopping power."
For example:
Speer's 9mm Standard Pressure 124-grain Gold Dot has a published velocity of 1150 feet per second (fps). Speer's 9mm +P 124-grain Gold Dot has a published velocity of 1220fps, a 70 fps increase.
Federal 9mm Standard Pressure 115-grain JHP has a published velocity of 1160 fps. The Federal 9mm +P+ 115-grain JHP has a published velocity of 1300 fps, an increase of 140 fps.
The downside to using +P or +P+ ammunition is the cost, a possible increase in perceived recoil and accelerated wear on the firearm.
FWIW, when I used a 9mm my load was either Speer 124-grain +P Gold Dots or Winchester 127-grain +P+ loads. While I often felt more recoil (sometimes I did not notice), my times between shots were just as fast as with standard pressure ammunition. I put several thousand rounds of both loads through my weapon without any noticeable effect.
IMO, when using 9mm or .38, it is desireable to use +P ammunition. That is not to say that adequate results can not be achieved with standard pressure ammunition.