What is the ideal gas equation and what relationships can we derive from it?

The ideal gas equation is given by pV = nRT, where p is pressure of the gas, V = volume, n = molar amount, T = temperature (in Kelvin!) and R is the gas constant which is given by 8.31 J / (K*mol). From this equation we can see that the pressure is directly proportional to the temperature and moles of the gas, so if one were to increase the temperature, pressure would subsequently increase (provided all the other variables are kept constant). This proportionality is the same for the volume. We can also see that the pressure and the volume are inversely proportional, meaning that an increase in one would cause a decrease in the other (again with everything else constant).