The first recipe I ever tried to make from a book was a cheese soufflé, and it called for separating the eggs. I think I was twelve. I was overly ambitious as a kid, what can I say?
At first I wondered, “What in heaven's name are you talking about, 'separating eggs?'” Separate them from what? The shell? After reading the rest of the recipe I got the idea. “Oh! Separate the yolk from the white! How the heck do I do that?”
My mom wasn't home so I called a neighbor, who was kind enough to come over an show me. I think the real reason she came over was to make sure I wasn't destroying the kitchen. She was my mom's friend, after all.
One problem with separating eggs is you don't always need both the whites and yolks in the same recipe. You could just throw the unused part out, but you don't have to. Egg whites freeze quite well for use in meringues and such, later on. Oddly enough, egg whites that have been frozen, and then thawed, whip up better than fresh ones.
The yolks are another story. They've got too much fat in them to freeze well and can easily freezer burn. Bacteria grows in them pretty easily, too, unless you can get the temperature below -20 degrees Fahrenheit. If you plan to keep the yolks, put them in the refrigerator under a layer of cold water. You can pour off the water before you use them. Even then, egg yolks won't keep beyond a couple of days. Unless I'm planning on using them by the next day, I either fry them up as a snack or throw them out. Salmonella is not your friend.
To separate the egg yolk from the white, crack the egg on the counter and open it over a bowl, keeping one half upright to hold the yolk. Let the white spill out into the bowl.
Gently pour the yolk into the other half, letting more of the white drop in the bowl as you transfer the yolk back and forth between the two halves. Don't do this more than a couple of times or you'll probably break the yolk. I think it's better to let a little of the white stay with the yolk than to let any yolk get into the whites. Even a drop of egg yolk in the egg whites will stop a meringue from whipping up nicely.
Pour the egg yolks into a separate bowl when you're done, and move on to the next egg.
If that seems a bit complicated, you can just pour the egg into your hand, catching the yolk and letting the whites run between your fingers. I prefer separating the eggs by pouring them between the egg shell halves, though. It's less messy.