I think the best way to explain gas pressure is using the "ping-pong ball" analogy.
Air, like any gas, consists of molecules moving around through empty space. For now, you can think of molecules as hard balls, like ping-pong balls, only much smaller. A container full of air is like a box full of bouncing ping-pong balls. The ping-pong balls are constantly moving, bouncing off each other and off the sides of the box.
When a ball bounces off the wall of the box, it pushes against the wall. The force produced by a single ball is tiny, but if the balls are moving fast, all the bouncing balls together add up to a large outward push on the wall. This outward push by the balls is the same as the "atmospheric pressure".
If we take the same box and put twice as many ping-pong balls in it, a ball will bounce off the wall twice as often as before; since each bounce provides the same amount of push, the wall will be pushed back twice as hard. This is why adding more air to a container (like an air tank) makes the pressure go up.
If we make the balls move faster, then they'll bounce into the walls more often, and with more force, so the wall will be pushed back harder. The speed of the balls is like the temperature of the air... but I'm afraid I can't explain why without using math. Thus, the pressure in a container also goes up when we heat up the air inside.
Try the links in the MadSci Library for more information on Earth Sciences.