Cream of tartar does not affect the stiffness to which egg whites can be beat nearly as much as it helps stop beaten egg whites from collapsing. 

When you beat egg whites, proteins in the whites unfold from their natural shape and become tangled with each other.  At the same time, you are beating air into the whites, forming small bubbles.  The protein molecules become attached to each other through chemical and electrical bonds that reinforce the skin of the air bubbles.  Over time, these bonds can pull the proteins closer together, forcing out the water trapped in the surface of the bubbles.  Eventually, the proteins pull themselves together so strongly that compact, grainy protein lumps form and the liquid pools in the bottom of the bowl.

This is where the cream of tartar comes in.  It helps prevent the formation of chemical bonds between protein molecules.

To understand this, first a little chemistry.  Protein molecules have, along their length, sulfur atoms that in turn may have a hydrogen atom attached.  If the sulfur atoms release their attached hydrogen into the surrounding liquid, then they can attach to something else.  Sometimes, that something else is a sulfur atom on another protein molecule.   If too many sulfur - sulfur bonds occur, the proteins pull together too strongly and the foam collapses, as above.

Because it is acidic, the cream of tartar increases the number of free hydrogen atoms in the liquid.  These extra hydrogen atoms either stop the sulfur atoms from releasing their attached hydrogen, or else provide another hydrogen atom to quickly take its place.  Because of this, the rate at which sulfur - sulfur bonds can be created is slowed down enough to help keep the egg whites stiff long enough to be used.

Any acid can be used for this purpose, although cream of tartar and lemon juice are the most common.  The amounts needed are 1/8 teaspoon per egg white for cream of tartar and 1/4 teaspoon per egg white for lemon juice.

If you are having trouble beating egg whites to stiff peaks, there may be one of three causes.  If you got any of the yolk from the eggs into the whites while separating them, this can lessen the ability of the eggs to form stiff peaks, as can contamination from oil, fat or detergent on the utensils.  You should never use a plastic bowl to beat egg whites, since fat molecules are attracted to some plastics and may leave a film on the inside of the bowl.