Egg whites can be beaten by hand or with an electric mixer. The main difference is how much personal effort you want to put into it vs. how much stuff you want to clean up afterward.
The bowl you use can make a lot of difference. According to Recipe Tips, copper is best because it is slightly acidic. The acid helps stabilize the foam as the whites are beaten.
The much cheaper stainless steel bowls can be used. The trick to using stainless steel bowls is to replace the acid without changing the flavor of the eggs. I pinch of cream of tartar will substitute for the acid in the copper bowl.