$PV = nRT$

$PV = nRT$ is more commonly known as the Ideal Gas Law. This law states that the product of pressure $P$ and volume $V$ is equal to the product of the amount of the gas in moles $n$, the ideal gas constant $R$ ($8.314*J*K^{-1}*mol^{?1}$), and the temperature $T$.

Intuitively, this makes sense. Assume that you have a fixed amount of gas inside of a cube. If you increase the pressure of the gas inside the cube, by definition, you are increasing the force per area of the gas. When you apply a force on something, you can say that you are transferring energy from yourself to whatever you are applying a force on. The transfer of energy will increase the overall energy of the system, causing the atoms in the gas to move faster and therefore increase in temperature. Therefore, increasing pressure and keeping volume and amount constant causes an increase in temperature, which is shown by the Ideal Gas Law.

Now say you keep the pressure constant and increase the volume of the cube that the gas is in. This is slightly harder to visualize, so we can turn to the units on the left side of the equation to help us out.

Unit of pressure: ${N}/{m^2}$

Unit of volume: $m^3$

Unit of pressure * volume: ${N}/{m^2} * {m^3} = N*m = Joule$

A joule is a unit of energy. So if we increase only the volume and keep pressure constant, the energy on the left side of the equation will increase. Using the same argument from the previous paragraph, an increase in energy of the system will cause the atoms of the gas to move faster and will cause the temperature of the system to increase. Therefore, increasing volume and keeping pressure and amount constant causes an increase in temperature, which is again shown by the Ideal Gas Law.

These relationships can be expressed through the equation (a simplified version of the Ideal Gas Law):

${(P_1V_1)}/{T_1} = {(P_2V_2)}/{T_2}$