With practice, I've figured out how to substitute eggs in a wide variety of baked goods without using specialty ingredients and without affecting the flavor or texture of the finished bake. Here's how to determine what eggs do in recipes, and my advice for how to swap them out:
What Eggs Do in Baking
Eggs add a few different qualities to baked goods. An extra-large egg contains about 1/4 cup of liquid, which will make your batter or dough moister. They also have a leavening effect, which means they make the batter or dough puff up in the oven. Large eggs have about 5 grams of fat, mostly in the yolk, which is equivalent to about 1 1/2 teaspoons of butter. They also have about 6 grams of protein, mostly in the white, which adds a chewy texture to homestyle baked goods like brownies and drop cookies. Finally, eggs have an emulsifying or binding effect that encourages mixed ingredients to stay together rather than to separate.
Substituting Eggs in Cakes, Muffins, and Quick Breads
In batters for cakes, muffins, and quick breads, you'll need to replace the moistening and leavening effects of eggs. Simply add 1/4 cup extra of whatever liquid is called for in the recipe per egg replaced--usually this will be milk. You can also use 1/4 cup extra per egg of applesauce, mashed banana, pumpkin, or any other fruit puree called for in the original recipe.
To replace the leavening, I find 1/2 teaspoon of baking powder per egg replaced works well in most cases.
Substituting Eggs in Cookies
In cookie doughs, you'll need to replace the moistening and texturizing effects of eggs. The key here is to avoid making the finished result too cakey, which I find can happen if milk is used in place of eggs. Instead, I like to replace some of the sugar in the recipe with a liquid sweetener, which adds moisture to the dough and imparts a chewy texture. To substitute, replace 1/4 cup of granulated or brown sugar with 1/4 cup of maple syrup or molasses per egg called for. (Molasses is darker and more bitter than maple syrup, so keep that in mind.)
Since eggs also bind fat into the cookie dough, leaving them out can result in butter or oil melting out of your cookies into a greasy puddle in the oven. Decreasing the total butter or oil used by 1/4 will prevent this--and make your cookies a little lighter to boot!
My Egg-Free Recipes
If you're looking for a recipe that's already free of eggs, I have several you could try:
Fat-Free Vegan Doughnuts
Lighter Egg-Free Chocolate Cupcakes and Vanilla Frosting with No Mixer
Egg-Free Lighter Chocolate Chip Cookies
Egg-Free Savory Cheese Muffins
Egg-Free Crispy Cocoa Cookies
Vegan Peanut Butter Cookies with No Added Shortening
Simple Vegan Chocolate Shortbread with No Added Shortening