The "stiff peak" phase of beating an egg is very important to the texture and volume of dishes such as meringues, some cakes, souffles, custards, mousses, and many more. To reach this phase, beat eggs until they reach the "soft peak" phase, where lifting your beating implement from the eggs will result in the formation of a softened peak. If the recipe calls for adding sugar, do so very gradually while you continue to beat the egg until raising your beater results in a glossy, stiff peak. Do not beat the eggs beyond this point.
So why do stiff peaks form?
The egg white (or albumen) contains a number of proteins, including ovalbumin, conalbumin, ovomucin, globulin, lysozyme, avidin, ovoinhibitor and flavoprotein. Beating egg white causes its protein to denature, or break apart at certain chemical interfaces (hydrogen bonds, etc). At the same time, the beating causes bubbles of air to enter the fluid and form a foam.
As beating continues, the air bubbles are continually redivided and form a thin elastic film with the denatured protein, increasing the volume of the mixture.
The ovalbumin proteins lend elasticity to the foam, allowing peaks to form, while the other important proteins aid in the stabilizing of the air bubbles and the creation of a stable foam.