To separate an egg, crack the egg on the edge of a bowl and open it up over a smaller bowl, holding the larger half-shell beneath the smaller one. Carefully let the yolk settle into the lower shell. As you do this, the white will spill out into the bowl below.
Transferring the yolk to the smaller half of the shell, let the rest of the white spill into the bowl.
Transfer the yolk one more time, then place the yolk into a separate bowl.
It's a good idea to separate eggs one at a time into a small bowl and then transfer the separated egg into a larger bowl. This way, if some shell or yolk should fall into the white, it is easier to remove it. This is important, because if you're going to whip the whites the fat from the yolk will prevent proper aeration.
If you have a lot of eggs to separate, try this: Simply pour the cracked egg into your hand and let the white slip between your fingers into the bowl. The yolk remains behind.