I've talked quite a bit about both egg yolks and egg whites, but I've never really talked about how to separate them from each other. It's time to take care of that.
In general terms, the egg has four separate components. It has a shell, a yolk, a white and a chalazae. The chalazae is a rope-like structure that attached the yolk to the white, and helps the yolk stay centered in the egg. I make a point of mentioning it here because it is not actually part of the yolk or the white, and you will encounter it when separating eggs.
Before we get into the heart of this tutorial, we need to get set up. You will need three non-reactive bowls, set up in a row. By non-reactive, I mean glass or stainless steel. Don't use aluminum or plastic bowls. With aluminum you run the risk of adding off colors and/or flavors to the eggs. Plastics are bad too because the molecular structure of plastic is too close to that of fats. They are so close in fact, that plastic often bonds to fats, and that bond can remain for several washings. If your egg whites are stored in a plastic bowl, they run the risk of picking up fats that were previously stored there, which will render them unsuitable for whipping later.
I also lay out a paper towel, or sometimes a sheet of wax paper, next to the bowls. When you crack eggs, some of the white will start to leak between where ever you cracked it and the bowl. This means a messy counter, especially with a lot of eggs. The paper towel makes it easier to clean up later, though you will still want to wipe the counter under the towel when you're done. The towel also gives me a place to store used egg shells while I'm working. When I'm done, I just wrap up the shells in the paper towel and carry them to the trash can.
One of the most important parts of separating an egg is cracking the shell properly. If you hit the shell too hard, it will shatter into pieces. If you don't hit it hard enough, it won't really open. It's something that takes practice, but you need to crack the egg just enough so that it will open into two halves, roughly the same size. I often will hit an egg lightly against the counter, and then rotate it in my hand and hit it again. After two or three hits, it generally opens pretty easily.
This is when we actually start separating. I have seen many a tool for this sort of thing, and I'm happy to tell you that you don't need any of them. The only tools you need are already right in front of you: your hands. Some people use the egg shells as tools, and that method is perfectly acceptable. I find it tedious, and so I stick with my hands. I will, however, cover both methods.
Hold the egg above the center bowl and remove the upper shell. Some of the white will immediately spill into the bowl below it. The goal is to get as much of the white into that center bowl as possible, without getting any of the yolk into it. The fresher the egg, the more easily the two will separate from each other. I mentioned the chalazae before. It doesn't really matter where that part goes, so don't worry about it. I mention it again because some people will wonder what the heck to do with it. Don't worry about it, it's nothing.
To separate the egg, just dump the contents of that bottom shell into my other hand, leaving small cracks between my fingers that the white can slip through, but not the yolk. If you're using the shell method, you will carefully pour the yolk back and forth between the two halves of the shell, being careful not to let the edges of the shell cut through the yolk. This is exactly why I don't like the shell method; your hands will likely have far fewer jagged edges (hopefully none). I've also found the shell method to take too long for my tastes. After a couple of passes, the white will usually fall into the center bowl, leaving nothing but yolk inside the shell. Inspect the yolk. If there is any blood in it (and sometimes there will be), throw the yolk out. Believe me, you don't want it. If it's clean, go ahead and dump the yolk into one of the side bowls and discard the shell.
This part is important. Inspect the white in the center bowl carefully. There should be no yolk in it. If even a speck of yolk made it to the center bowl, discard the white and clean the bowl using soap and hot water. Even a speck of yolk will render egg whites unsuitable for whipping. If you notice while you're separating that the yolk has broken, dump anything that's left of it into your yolk bowl. Any remaining whites will do far less damage to the yolks than the yolks will do to the whites. Also, inspect the white for blood. If there is any, throw the white out. Once you've decided that the white in the center bowl is clean, go ahead and dump it into the other bowl that doesn't contain yolks.
At this point, you will have a separated egg! Many bakers will actually separate two at a time into the middle bowl, before dumping it into the whites bowl. You can just stick with one at a time until you get practiced. Most (but not all) recipes that call for separated eggs will call for many, so go ahead and separate the others. Keep the yolks in one bowl and the whites in another. If ever a piece of yolk gets into the center bowl, make sure to wash it before continuing. And yes, soap is the only way to be absolutely certain that there is no more fat in the bowl.
Did your recipe only call for egg yolks? If you're making ice cream or cheesecake, it might have. That leaves you with a whole mess of egg whites. These can be frozen for up to a month, maybe even two, and then thawed when you're ready for them. I've seen many a cook freeze them in ice cube containers, and once frozen they move the cubes to a zip-top bag (always with the date on it). This method is fine by me, so long as the ice cube tray has never been used to store anything with fat in it (you should probably be okay there). To thaw, move the whites to the refrigerator for a day. This should give them plenty of time to thaw properly and safely.
Did your recipe only call for egg whites? If you're making angel food cake or meringue, it probably did. If you're planning on putting together such a recipe, I would suggest finding another one that uses yolks, and making that at the same time. I have safely stored yolks in the refrigerator overnight, but I don't trust them after that. If you can't use them within 24 hours, throw them out. They don't store nearly as well as the whites.
Hopefully I have given you enough information to properly separate eggs. It may look intimidating, but if you do it enough, it will become second nature to you. Go ahead and whip up an angel food cake or a batch of ice cream. Get some practice separating some eggs. It's not nearly as complex as you'd think, and it's an extremely valuable skill to have.