When you apply for a job, there is an interview. Your potential employer wants to get an idea of how you will work out for their company, and in most jobs people skills are a must. In the culinary world though, people skills don't count for much. The real talent is how well you cook, and the omelet is your interview.
So why the omelet? They are deceivingly hard to do right. Anyone can make a mediocre one, but a good (or great) one takes skill.
What defines a good omelet first of all is the look. The bottoms should be uniformly yellow, with no brown spots from overcooking, and smooth. The inside should be very soft, almost but not quite undercooked.
There is no taking your eyes off of an omelet as it cooks, nor should your hands be idle. It's a fast process too, so you need a pre-heated pan and moderately high heat. A non-stick pan is a must in this case as well.
When the egg is added it needs to be swirled around the pan quickly to evenly distribute it. You don't flip an omelet typically, so to get everything cooked you pull the edged back and tip the pan so the raw egg runs underneath.
The filling is usually cooked before hand, and adding it before folding over the omelet. There should be enough heat left over to melt any cheese, but not overcook the eggs.
All this happens in under a minute.
The omelet test several skills at once. First of all it shows how educated you are. Someone who has been through culinary school will cook it differently than someone who has not. The chef knows how much instruction is required at this point.
It tests how observant you are. Every stove is slightly different, so you need to know how to adapt. To make an omelet you need to watch your heat, how fast the eggs set, and when to add your filling. It tests how well you can time things, and how fast you can adjust as you go along.
How you treat the omelet tells the chef how you might treat other food. Do you do it with speed, recklessness, or take your time? The last one is not always a good thing by the way. Do you take care as you do it, or are you not as observant as you should be?
Also, it's a cheap way to test someone's skills. Eggs cost well under a dollar, and a basic omelet is usually just filled with cheese. If you flub the test, the total cost is under a dollar. Try doing that with a T-bone steak. That mistake costs closer to $20.
Now I have a confession to make. I never mastered the omelet test in my brief foray into the culinary world. My omelets always came out wrinkly on the bottom, though the rest of it came out fine. I needed more practice on twirling around the egg when it hit the pan. Five years later I'm pretty out of practice too, so I doubt I could make a good one on my first try (I don't like them myself).
The skills though are more important then the actually omelet anyway. Making good food is about paying attention, and knowing how it should look, feel and sound before it ever hits the plate. It's also about practice and doing it wrong on occasion, learning from your mistakes, and making it better next time.