How do different air pressure affect the weather?
It's more like the weather affects the air pressure.
Air pressure is changed by two things in the atmosphere, heat and moisture. According to Gay-Lussac's law, heat and temperature are proportionate. Therefore as heat increase temperature increase.
The atmosphere is primarily made of Nitrogen and Oxygen, everything else is in very small amounts. As nitrogen and oxygen are both diatomic molecules (2 atoms each) both those molecules have more mass than a molecule of water. This means that water vapor is lighter than air. We can see this when we notice that steam (water vapor) rises in the air. Therefore pressure will decrease with more water vapor.
So since weather will bring changes to both heat and moisture it is a little more correct to say that weather affects air pressure than to say that air pressure affects the weather.