The relationship between the two is that air temperature changes the air pressure. For example, as the air warms up the molecules in the air become more active and they use up more individual space even though there is the same number of molecules. This causes an increase in the air pressure.
The same goes for cooler air. When the temperature cools, the molecules slow down and they do not move and bump into each other. This causes a decrease in air pressure.
An easy way to understand this concept is by comparing car tires in the winter and car tires in the summer. In the summer the air is a lot warmer so the molecules are moving around a lot. The tire expands and you not need as much air because the pressure in the tires is high enough. In the winter, when the air is cold, the molecules are moving very slowly. Many people need to add more air to their tires because there is not enough pressure.
As temperature rises, so too does pressure. As pressure rises, temperature rises along with it. Obviously the opposite is also true if they are decreasing.
One great example of this occurs if you happen to use a propane tank (particularly the small ones) while cooking at a campout or picnic. As the gas inside the tank is burned and the pressure falls, you will very often see frost form on the outside of the bottle if it isn't too hot out. As the pressure inside has fallen precipitously, so too will the temperature on the outside of the bottle, leaving you with the frost.
Check out Gay-Lussac's law about the direct relationship of temperature and pressure.
Pressure and temperature are related to each other by Gay-Lussac's Law. This law says that P1T1=P2T2, or the initial pressure multiplied by the initial temperature is equal to the product of the final temperature and the final pressure. Pressure and temperature have an inverse relationship. In other words, increasing one, will cause the other to decrease.
So if you increase the air temperature, the pressure is going to decrease. If you decrease the air temperature the pressure is going to increase.
Before answering this question, it will be helpful to clarify some terms. When we say air, it generally refers to the mixture of gases that is present in the earths atmosphere. This should no be mixed up with gases in general. Also we need to realize that the air as it exists naturally in the atmosphere is impacted by the uncontrolled influence of the environment in which it exists. However, the conditions of air in a closed space can be controlled by an experimenter.
Now coming to the main question. Air being a mixture of gases, behaves in line with various laws of gases. One of this laws states that all other things being equal, the pressure of a gas increases with increases in temperature. This given a fixed quantity of air, contained in a space with fixed volume, its pressure will increase with temperature. However as the air exists naturally in atmosphere, there is no constraint on change of its volume. Therefore, the air will expand with increasing temperature. This will cause the the density of air to reduce. This in turn will cause the hot air to rise up in atmosphere. This upward movement of air actually tends to create partial vacuum and thus reduce the air pressure at places which are hotter. To fill this region of partial vacuum or low pressure cooler air from adjacent regions will flow towards it.
They work with each other while building and up holding each other up while increasing