Air pressure is dependent on a few things.  First is gravity.  Air is made of matter.  Matter has mass.  Mass exerts gravity and is pulled by gravity; therefore, air has mass, is pulled by gravity, and has weight. That means it exerts a downward pressure.  Air also exerts pressure in all of the other directions too, because it is a fluid.  That pressure is affected by the fluid's density and temperature.  Denser, colder air means more pressure.  The last general item that affects air pressure is depth.  Think of the atmosphere like a really big swimming pool.  The deeper you go, the more pressure is exerted.  

Okay enough general gas stuff.  Let's apply it to the atmosphere and weather.  An air mass is a big chunk of air over a particular area.  It will have fairly uniform temperature and humidity within it.  Air masses can form over land or over water.  Land air masses are called continental and water air masses are called maritime.  Those two categories give a rough estimate of humidity.  Continental tend to have low humidity and maritime tend to have high humidity.  Next is temperature.  An air mass can be warm, or it can be cold.  In weather jargon that's tropical or polar.  Combine those terms to get air mass characteristics.  Continental tropical is warm and dry (think Arizona).  Continental polar is cold and dry (Canada).  Maritime polar is wet and cold (New England area), and maritime tropical is wet and warm (Florida).  

Air masses have edges.  If they didn't, a single air mass would cover Earth. Those edges are called "fronts."  Fronts bump into each other, dive under one another, ride over one another, etc.  The reason they do that is because they have different humidity and temperature.  In other words, they have different pressures.  Those different pressures and temperatures don't mix well, so the air along a front is often more turbulent.  Meteorologists can track air masses and fronts by looking at barometric pressures.  Your question stated that air pressure is a super important indicator of current weather and incoming weather.  Absolutely.  As barometric pressure rises or falls, it indicates what kind of front is moving in and what type of air mass is behind it.  

There are four types of fronts.  

1. Warm front.  Indicated by a slightly falling barometer.  What's happening is that a warm air mass is approaching and over riding a cold air mass.  The warm air mass rides and up and on top of the cold air.  That slightly rising air will drop the pressure.  Then the cold air is sort of squeezed out from underneath and the barometer will rise because the warm air settles back down.  At this point nice, stable, generally warm air is in the area.  Go outside and throw a Frisbee.

2. Cold front.  Cold weather is coming in.  Once the cold air mass is over an area, there will be a rise in the barometer.  That's because the air is dense.  At the front however, the barometric pressure will fall a little bit.  That's because as the cold air comes in, it scoops out the warm air and pushes it up and out.  That quickly rising air causes air pressure to drop.  That quickly rising air also condenses quite quickly and forms storm clouds.  If a thunderstorm is going to develop, it's likely to happen with a cold front. After that warm air is gone, the pressure will rise indicating that the cold front is over the area. 

3. Stationary front. Neither air mass has sufficient movement or pressure differences to push the other out of the way. Conditions are generally stable and similar to a warm front. Barometric pressure won't change much, if at all.  

4. Occluded front.  You need 3 air masses for this one.  Two cold air masses and a warm air mass in between.  I'll line them up.  On the right, place a receding cold air mass.  Just to the left of that, place a warm air mass that is trying to push out and overtake the cold air to the right.  On the far left, another cold air mass, but it is moving toward the other two quickly.  What this does is "pinch" the warm air in between and force it up and out.  

All of those fronts and masses can be tracked using air pressure, so it is an important weather tool.  However, seeing a falling or rising barometer doesn't guarantee a certain type of weather.  It just allows meteorologists to more accurately predict what they believe is coming.