The method stated in textbooks and such say that:
The separation of a gas-gas mixture (e.g. Nitrogen + Oxygen) can be performed through a process called liquefication. First the two gases are placed into a "special chamber" where they are cooled to extreme negative temperatures (in the case of oxygen and nitrogen, to at least -200C). Once the gases have been frozen, they are then liquefied/melted. The temperate in the chamber is increased until one of the gases turns into its liquid form.
Once the liquefication process is complete; the gases in their liquid state are then taken through Fractional Distillation (another seperate process you may have read about), where the two or numerous liquids will go through many complex evaporation-condensation process until the liquids are able to be split. In the case of the mixed nitrogen and oxygen, you will end with pure substances (with maybe very minor traces of the other mixed element(s)).