Great question! Air in the atmosphere is subject to a wide variety of temperatures at different parts of the day and at various heights. These changing conditions can drastically affect the density of air, which in turn affects things like humidity capacity and are a large part of what causes different weather systems (think about how cold air sinks and hot air rises, and how that can lead to vortexes or air cycles). Simply, the air density can be approximated with the Ideal Gas Law you may have learned in chemistry:

Where P is the absolute pressure, V is the volume of gas, n is the moles of gas, R is the gas constant, and T is the absolute temperature. The density ρ is calculated by dividing the amount of gas by the volume of gas. This leads to the equation:
ρ=P/(R_s T)

At sea level and about 15 °C (59 °F), the density of dry air is about 1.225 kg/m³, but temperature can vary this value. For example at 35 °C (95 °F) the density is 1.146 kg/m³, and at -25 °C (-13 °F) the density is 1.422 kg/m³.

When air cools (assuming everything else is held constant), the density increases. If you're in chemistry, you will eventually see ideal gas law and where it comes from. When air is heated, it expands. This is why hot air balloons fly: the warm air is less dense than cooler air and the balloon floats. Also, combustion engines (in cars and other vehicles) take advantage of this expansion of cool air to move a piston. However, it is important to note that air density is also affected by surrounding pressure (warmer air will not always be less dense). If a gas such as air is under less pressure, it will expand and become less dense. An example of this would be cities like Denver, Colorado. It is in the mountains. Even though the weather there is cold, the air is less dense than air here in Santa Barbara because the pressure is lower up in the mountains. Therefore, the air can expand more. (The pressure is lower in the mountains for the same reason that pressure is higher in the deeper parts of pools and lower in the shallower parts. There is more water pressing down on top of the deeper parts, just as there is more air above Santa Barbara pressing downward than there is above Denver because it is higher up.)

Good question! Here's a related question: why do hot air balloons float? Well, in order to float, the air inside the hot air balloon has to be less dense than the air outside of it (that's the same reason air bubbles will rise when they're underwater). So hot air is less dense than cooler air.

So that's the short answer to your question: when the air cools down, it becomes more dense. But wait - denser air is heavier, so it pushes down on the ground more. That's why colder weather is usually associated with high pressure weather systems. Similarly, since warmer air rises and is less dense, warmer weather is associated with low pressure systems. And since things tend to want to move from high pressure to low pressure, air will tend to move from high pressure (cold) weather systems to low pressure (warm) weather systems. That's what causes wind and weather!

Initially, density goes up because it's easier to pack more molecules into a smaller volume when the temperature is lower. However, pressure remains roughly the same, because any inequality in pressure results in air flowing from high-pressure areas to low-pressure areas (this is what causes wind).