In order to focus the eyes, the muscles of the eyes are constantly making small movements to adjust. Normally, these movements are so small and fast that they are not visible to others. Drugs and alcohol cause muscle response to slow down and muscles to become less coordinated.

In the eyes, this slowed response and lack of coordination shows itself when a person is asked to follow a pen back and forth - the normally invisible small and fast movements become visible as slow and jerky movements. The more impairment there is, the more the eye muscles jerk and twitch. This jerking and twitching is called nystagmus.

Thanks to Phoenix Traffic Officer Scott for his help answering this question.