Air pressure and density primarily affect the physics or dynamic movements of air. This includes rising warm air and sinking cold air. As you can see in the list below, air pressure and density affects weather by controlling the flow of air, the transportation of heat and moisture and cooling and warming of he atmosphere through expansional cooling and compressional warming.
1. Air Pressure differences affect the flow of air (wind). Because air naturally flows from areas of high pressure to low pressure, pressure differences over an area (pressure gradient) causes air flow or wind. Air pressure is the main cause of localized and planetary scale wind.
2. Air Pressure affects the transportation of heat and moisture. Wind, which is caused by air pressure, is able to transport heat and moisture from one place to another. For instance, air blowing from warmer sea surface temperatures will likely transport heat and moisture. In contrast, cold arctic and antarctic air would likely bring cold temperatures.
3. Adiabatic cooling and warming. Air pressure affects temperature through adiabatic processes. In low pressure systems, a warm air parcel rises, expands, cools and then condenses. This is why areas of low pressure are typically associated with rain. In contrast, in high pressure systems a warm air parcel, sinks, shrinks, warms and dries up. This is why areas of high pressure are often associated with warm and sunny weather.