Here is the way to perfectly whip egg whites. A copper bowl is ideal, because copper reacts with the albumin in egg whites and makes for better foaming.

Wash your bowl with vinegar and kosher salt, then rinse well just before adding the whites. Use a whisk or an electric hand mixer on a low setting until the whites are foamy. At this point, add a pinch of cream of tartar to stabilize the whites. This helps them to hold their shape longer. Continue to whip. If using a hand mixer, increase the speed. If your recipe calls for sugar, add it as the whites are beginning to form soft peaks. Whip the whites until they form stiff, glossy peaks that retain their shape. If you over-whip unsweetened egg whites, they will lose their glossy appearance and their ability to stay in firm peaks.

Notes
A large balloon whisk is the easiest way of getting air into the egg whites. Egg whites should expand to four times their normal size during the whisking process.

A copper bowl is the best kind of container for the whipping, too, because copper strengthens the protein of the whites, giving them more height and stability.