The pressure of the air is a good indicator of the current stablity or instabilty of the atmosphere. In other words, knowing the pressure---and whether it's rising or falling---can tell you a lot about what change may be coming next in the weather.
Rising pressure generally means sunny and drier weather is moving in. Falling pressure generally indicates more clouds and possible precipitation. Also, the actual pressure reading can be an indicator of just how sunny or how stormy things are likely to be. A key statistic we look for when forecasting the strength of storms, for example, is the central pressure, or the lowest pressure expected at the center of the storm. The lower this central pressure, the stronger the storm and the worse off we may be. Similarly, a center of high pressure with a relatively large reading can mean exceptionally tranquil weather and perhaps a hot spell, depending on the season.