You're making a butter cake from scratch. Your cake pans have been buttered, greased, and set aside. The oven is preheating, and you've nearly finished creaming the butter and sugar. It's almost time to add eggs to the batter (for most traditional butter or oil based cakes, at least), and the recipe is telling you to do it one egg at a time, with ample mixing between additions, using eggs that have been brought to room temperature. You're wondering, "Why all this fuss when adding eggs to cake batter? Why can't I just pull eggs out of the fridge, crack them, and throw them into the mixer?"
All of those extra little precautions we take (creaming butter and sugar completely, allowing eggs to come to room temperature before adding them slowly, mixing vigorously and thoroughly between each addition, obsessively scraping the sides and bottom of the bowl, plus the beater...) are in the service of creating a stable emulsion of liquid and fat in the batter. When this is done correctly, the cake will have a springy, even crumb, great flavor, and light texture. If the batter is not properly emulsified, the resulting cake can be uneven and flat, flavorless, and have a heavy texture. It's even possible for the cake to "fall" or sink into itself when baking.
An emulsion is the suspension of small globules of one liquid in a second liquid with which the first will not mix, such as oil and vinegar. To create an emulsion, we slowly add eggs to the butter mixture, one at a time, beating rapidly to suspend the water from the butter and egg whites in the fats from the butter and egg yolks. Lecithin, an emulsifier found in eggs, helps to stabilize the emulsion, as does aeration from steady, rapid beating.
Adding the eggs slowly makes it easier disperse the liquid, and promote even suspension of liquid in fat. Ensuring that all of the ingredients are around 70 degrees Fahrenheit will also help them combine more easily and smoothly by promoting aeration in the egg whites and preventing curdling. If your emulsion breaks (looks separated and weepy), do not worry, there are easy ways to fix it and move on.
Click through this slideshow to learn step-by-step tips and tricks for preparing your eggs for mixing, adding them to cake batter, and solutions for fixing broken emulsions. Then click over to this recipe for Gale Gand's amazing Chocolate-Almond Upside-Down Cake, and enjoy a delicious practice session of your own.