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Air is composed of molecules, all of which have a mass. While you may think this mass would be small, the accumulation of all these molecules between you and the edge of the atmophere actually exerts about 1 kg per square cm (14.7 pounds per square inch) of pressure, weighing down on you, constantly!

Air pressure also comes from the movement of molecules. They bounce around and push against each other; the more molecules there are (or the denser the air), the more they push, and the more you feel that push on you. This is why denser air is warmer - there's more bounce, more friction, more heat.