High air pressure is when the atmosphere's pressure is above average. This is due to a simple wind pattern, which is distorted to such a degree that the air is twisted to the right or left. Great wind spirals are produced, which we know as high and low pressure systems.
In the Northern Hemisphere, the air in high pressure systems spirals clockwise and outward from the center. In the Southern Hemisphere the direction of the spiraling air is reversed.
High pressure systems are domes of density that press downward. With the increasing air pressure, the temperature rises; underneath those high pressure domes, the air tends to sink into the lower levels of the atmosphere where temperatures are warmer and can hold more water vapor. Any droplets that might lead to the formation of clouds generally evaporate. The end result tends to be a clearer and drier weather pattern.