The atoms and molecules that make up the various layers in the atmosphere are constantly moving in random directions. Despite their tiny size, when they strike a surface they exert a force on that surface in what we observe as pressure.
Each molecule is too small to feel and only exerts a tiny bit of force. However, when we sum the total forces from the large number of molecules that strike a surface each moment, then the total observed pressure can be considerable.
Air pressure can be increased (or decreased) one of two ways. First, simply adding molecules to any particular container will increase the pressure. A larger number of molecules in any particular container will increase the number of collisions with the container's boundary which is observed as an increase in pressure.
A good example of this is adding (or subtracting) air in an automobile tire. By adding air, the number of molecules increase as well the total number of the collisions with the tire's inner boundary. The increased number of collisions forces the tire's pressure increase to expand in size.
The second way of increasing (or decreasing) is by the addition (or subtraction) of heat. Adding heat to any particular container can transfer energy to air molecules. The molecules therefore move with increased velocity striking the container's boundary with greater force and is observed as an increase in pressure.
