In the DWI context, intoxicants, according to the NHTSA, can increase nystagmus or the amount of involuntary eye-jerking. Alcohol consumption or consumption of other "central nervous system depressants" hinders the ability to control eye muscles correctly. This, therefore, causes the jerking or bouncing of the eyes.
Alcohol can cause two types of nystagmus (1) alcohol gaze nystagmus (this includes HGN) and (2) positional alcohol nystagmus. When a police officer is conducting tests, they are not testing for positional alcohol nystagmus.
In broad terms, gaze nystagmus is a jerking of the eyes where the eye gazing upon or following an object (e.g., pen, pencil, stick). What happens is that the eye begins to lag behind, then it "jerks" to catch up. This will happen several times.
The test is generally administered when an officer holds a pen or pencil in front of your face. The officer then will move it to the right or left, in a slow, controlled fashion. The officer will then look to see if your eyes jerk when the eye is following the pen.
The alcohol that a person allegedly consumes results in poor motor coordination and slow reflexes. Thus, the nervous system, which controls eye movements, cannot cause the eye to roll smoothly; instead, it lags behind and jerks. At least, that is what the police will say. After all, there is generally not a video of someone taking the HGN to show the jerking of they eyes that they are going to test for.