The trick to separating eggs is to remove the yolk from the white without breaking the former into the latter. Passing the yolk from shell to shell is a popular method, but moving a yolk between two sharp-edged objects is not the best way to keep it whole. First, crack the egg against a flat surface (edges of bowls and counters tend to push shards of eggshell inwards, threatening the yolk). Then pass the yolk between your hands, letting the white fall between your fingers and into a bowl.
If you need a lot of separated eggs (angel food cake, anyone?), and worry about one broken egg spoiling the whole batch, use three small bowls: one to crack the eggs into, one to place the yolks in after separating, and one to hold the whites.