When you beat an egg, you are essentially breaking the egg's membrane and mixing the egg's contents - the yolk and the white. Beating also adds air to the mix, which makes beaten eggs lighter in color and texture. To beat an egg, simply use a fork or an electric mixer to whisk the egg vigorously until the yolk and white are combined and the mixture is light and airy.
Adding vanilla extract to the mix is a matter of personal preference, but many find that it enhances the flavor of the eggs. If you are using vanilla extract, add it after the egg has been beaten.