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Summary

Forces cause acceleration. When there are no forces, or when all the forces cancel each other, there is no acceleration: air in motion will remain in motion, and air at rest will remain at rest.

The primary forces in the vertical direction are gravity/centrifugal acting downward and the pressure gradient force acting upward.

The primary forces in the horizontal direction are the pressure gradient force, friction, and the coriolis force.

The coriolis force, caused by the rotation of the Earth, causes moving objects to try to curve to the right in the Northern Hemisphere and to the left in the Southern Hemisphere.

In general, especially for the larger-scale weather systems, all of the forces are nearly in balance and the acceleration of the air is very weak.