When we measure for forecast air pressure we are normally doing this relative to a certain height and most commonly relative to Mean Sea Level. The air above us is made of consituents like nitrogen, oxygen, CO2, water vapour and many other molecules. Each of these constituent parts of our atmosphere have a weight associated with them and so the more of them that we pile up above us, the heavier the area. It is this weight of the air above us that we measure when we take a pressure reading with a barometer. The density of the air above us varies considerably and this has a direct impact on the weight. Variations in pressure from one place to another generate a pressure gradient, for example between a high and low pressure system. Air will flow down the gradient from high to low and the steeper the gradient the faster the flow ( this is what we call wind). It would be easy to imagine the wind flowing in a straight line between high and low but the air flow/wind is deflected by friction with the ground, the movement of the pressure systems and the rotation of the earth so that it looks at though it flows at nearly 90 degrees to the downhill gradient.
Some of the strongest winds in the world are associated with intense pressure gradients around hurricanes and tornadoes.
If you see the pressure either in observations or in the forecast falling or rising rapidly, then it is likely there will be strong winds nearby accompanied by possible significant changes in the weather and wind direction over the relatively near future (hours as opposed to days).
One of the simplest ways to see air pressure in action is with a straw and a glass of water. Start to drink by lowering the pressure in your mouth (sucking) and the outside pressure on the water in the glass will force the water up the straw to your mouth.