system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: When youre intoxicated, why do your eyes stutter back and forth when following a pen during a field sobriety test?
gpt
I don't know exactly how much info y'all want, so I'll do a tl;dr at the end. 

Brain signals that initiate muscle movement start in the cerebral cortex and move through the spinal cord to the muscles. While the nerve signals pass through this system ( specifically when they're in the medulla) they are molded by signals from the cerebellum. 

So basically(generalizing here)  the cerebellum controls fine muscle movement.  Unfortunately, drinking alcohol increases ~~production~~ (correction: uptake) of GABA (a neurotransmitter which primarily has a sedative effect) in the brain.  When there is an overabundance of GABA, the cerebellum doesn't function properly, and causes jerky, shaky fine motor movement.  Your eyes are controlled by fine motor movement, and thus get the same effect.

This is also why you have a hard time touching your nose when you are drunk.

TL;DR
When you drink, the motor control center in your brain gets too much of a specific neurotransmitter, and does not function well.


Disclaimer:  the GABA inhibitor theory is only that, a theory.  Alcohol may very well affect the cerebellum via different mechanisms.  However, this is a widely accepted theory and is the best one we have right now.