The purple glow in auroras is mostly due to electrons running into nitrogen (N2) and exciting it. It then glows in primarily blue and red, making the purple color.

The green glow in auroras comes from a specific quantum "line" from oxygen (O). The deep red, found above the green is also from oxygen (O) but from a different quantum line.

What color depends on how fast the electrons from space are entering the upper atmosphere. The really fast ones come down from space, pass by the oxygen, and hit the nitrogen -- making purple light. The slower ones never make it to the nitrogen before they loose all their energy running into the oxygen (making green and red light).

To make things more complicated, the oxygen lines (red and green) come from excited oxygen that stay excited for a long time (long "lifetime") before relaxing and glowing in red and green. So, sometimes the excited oxygen will bump into another atom or molecule in the air and lose its energy before it glows. This process is called "quenching." Since the red line excitation lifetime is 110 seconds long before it relaxes and glows compared with 0.7 seconds for the green line and about a nanosecond for the nitrogen lines, the collisions in the air also contribute to the color since some colors will get quenched as described.