You can get them in non-stick, aluminum, cast iron, enamelled cast iron, and stainless steel.
You can also get folding Omelet Pans. They consist of two half rounds, hinged in the middle. You heat it on the burner, and fill each side with egg mixture, and let it cook a bit. Then, scatter any filling ingredients on top of one side or the other, and cook more. Then, fold the pan to bring the two halves together, making an omelet with a filling in it. It creates half-moon shaped omelets.
Purists are scandalized by cooking an omelet in such a folding pan; they say it makes an egg sandwich, not an omelet.