A front is the name given to the boundary between two air masses. A cold-front is a region where cold air will replace warmer air. The two bodies of air do not mix, as different temperatures depict different pressures and densities (approximations can be achieved using the Ideal Gas Law (`P=nRT` ).
These densities and pressures are vital to forecasting weather.
For example, when a cold front moves through an area, the air moving in is colder. This cold air is also denser (by definition), so the hot air is pushed upwards - see graphic from ECN, link below. If the warmer air is humid enough, it will form clouds and may cause rainfall.
Warm fronts (which replace current air with a warmer airmass) often cause clouds and storms.
In general, warm (and therefore usually moist) air is unstable, compared to cold (dry) air.
This video is a 12 min overview of the different types of fronts (warm, cold, but also occluded and stationary) and how they relate to weather forecasting.