To a very good approximation, the atmosphere obeys the ideal gas law, which says:
p = (n/V)*R*T
p is the pressure
n is the number of moles in a sample
V is the volume of the sample
T is the temperature (in kelvins)
R is the universal gas constant.
The quantity n/V is the molar density. A sample of n moles of air has a mass of m = n*M, where M is the average molecular mass of the air molecules, so n = m/M. Plugging this into the previous equation:
p = (m/V)*R*T/M
m/V is simply the mass density, ρ
p = (R/M)ρ*T
R and M are both constants, so R/M is also a constant.
This equation says that pressure is proportional to the temperature at constant density, and pressure is proportional to density at constant temperature.
We can rearrange this to solve for ρ:
ρ = (M/R)*p/T
This says that density is proportional to pressure at constant temperature, and density is inversely proportional to temperature at constant pressure. If both temperature and pressure decrease, then you have to know by how much in order to determine whether the density increases or decreases; what's important is whether the ratio p/T after the change is greater (in which case the density increases), or less (in which case the densify decreases) than the value of that ratio before the change.
The ideal gas law is what's called an "equation of state". It relates the temperature, density, and volume of a system. If you specify the value of any two of these parameters, the value of the third one is determined.