When a colder air mass moves toward a warmer air mass: it rises over the warmer air rises leaving cooler temperatures in the region.
On a weather map, a blue line with triangles pointing in the direction the cold air is moving represents the front.
In a cold front, the colder air mass pushes under the warmer air mass, forcing it to rise. As the colder air mass moves, it doesn't simply sit on top of the warmer air. Instead, it rises over it, creating a wedge effect.
As the colder air mass moves through, it replaces the warm air, leaving cooler temperatures in its wake. The cold air displaces the warm air, so the cooler temperatures are left behind the front, where the blue line with triangles is located.