|Image from NASA|
Pressure is force exerted over a given area. In the atmosphere, the molecules in the air apply pressure to everything on earth, including us. For instance, individual molecules in the air push against tiny areas on the top of our head. The force that air exerts is called air pressure. The more air molecules there are above you, the greater the force they exert, so the greater the pressure.
Pressure is important because it is related to volume, density, and temperature. In the atmosphere, warm surfaces can heat the air above them, causing the air to become less dense and to rise. This can eventually result in clouds and precipitation in the areas of rising motion, such as in the center of low pressure systems. High pressure in the atmosphere causes the air to compress and sink, leading to clear skies and calm conditions.
Say there is a cold air column next to a warm air column, but they both have the same air pressure, the cold air column is denser, so it will have a smaller volume and be shorter than the warm air column. This explains why the tropopause is higher over warm areas like the tropics and lower over cold areas like the poles. In an attempt to equalize temperature, volume, and pressure, air is transported from the warmer column to the cooler column. This causes an initial rise in air pressure in the cold column and a decrease in air pressure over the warm column. Once the temperatures are equal, the heights, volumes, and pressures will become equal, too. This transport of air drives weather.