Simple pressure is (by definition) just a force per unit area. Suppose you have a box and you put it in the air. Then the air pushes in on the box. It pushes in on each side with a force equal to the pressure times the area of the side, and with the direction of the force perpendicular to the side.
If there is also air inside the box, it provides its own pressure that counteracts the atmosphere, but if there is nothing inside the box, the box will likely collapse. You can demonstrate this with an empty soda can with just a bit of water at the bottom. If you heat the can over a flame so that the can is full of water vapor, then flip it upside down into cold water, the can will be crushed by the atmosphere. The vapor inside the can condenses quickly and the pressure inside the can drops. The pressure outside from the atmosphere stays just as strong, and the can's material itself is not strong enough counteract the air pressure outside, so the can crushes down.
Return to the box sitting in the air. Suppose the pressure is somewhat higher at the front of the box than at the back. Then the force on the front is greater than the force on the back. The box will be accelerated backwards - the direction where more force is pushing. (Note that pressure on the front of the box pushes the box backwards, just as if you push on the front of your car, you are pushing it backwards.)
A common situation in which this pressure difference occurs is a fluid in a gravitational field, such as the atmosphere or the ocean. In that case, the fluid itself is held up by the pressure difference. This is called hydrostatic equilibrium. We can consider an imaginary box made out of just the fluid itself. There is pressure on all sides of it, but a bit more on bottom than on top, so there is a net force upwards. On the other hand, gravity has a net force downwards, and the two just cancel.
Imagine a long line of such boxes under the ocean. The pressure underneath each box is always a little bit greater as you go further down, so the pressure builds up. The pressure is just the weight of a column of water divided by the area of the column (plus the pressure at the surface.) Deep underwater, this pressure is very high.
When you have the pressure gradient pushing sideways rather than up/down, there is nothing to counteract the force, so we don't have an equilibrium any more. In this case, the fluid will move. In general, fast-moving fluid has low pressure and slow-moving fluid has higher pressure because as you go from slow to fast, you're accelerated, so you must have high pressure behind you and low pressure in front of you. However, this is only a heuristic, and the actual pressure depends on other factors, such as the role of friction, temperature, etc.