If you, like me, watch "Good Eats" like there is no tomorrow, then you would appreciate the fact that cooking in essence is about science. Its about reactions, its about bonding of different ingredients to come together as a palatable whole. This is most evident (and applicable) when it comes to Baking. It is at times important to understand why we use certain ingredients and what their role is. Specifically, let us look at eggs.

You might also find these helpful:
1. How to find out what the role of Egg is in your recipe?
2. Common Egg Substitutes in Cooking and Baking



Depending on its role, you can use appropriate substitutes to eggs without compromising (to a large extent) on the taste. To my knowledge, the various roles of an egg (and the appropriate substitutions) are:

To act as a binder (Coagulate)
The egg proteins act as a binder by holding the ingredients together (or as they say "coagulating") while cooking or baking. When they are beaten/ heated, they change the liquid mixture to a solid state. They help in preventing crumbling and provide structure for foods.
Examples:
Meatloaves, casseroles, burgers, quick breads, cornbread etc

Substitutes:
-In such cases, to substitute eggs you would need an ingredient that plays the same role. Like mashed potatoes, cooked rice, bread crumbs, cooked pasta, oatmeal, grated cheese, flaxseeds, commercial egg replacement products, Tahini, nut butters, silken tofu, tomato paste, arrowroot powder etc.
-In cases of Sweet based goodies (like drop cookies), you can use fruits like Banana, applesauce, Xanthan Gum, agar agar (or non vegetarian equivalent like Gelatin), custard powder, cornstarch to perform the binding function. Keep in mind that using fruits can alter both the density and taste of the finished product.
-In case of Baked Goods, you can use flaxseed (1 tbsp ground flaxseed with 3 tbsp hot water, whisked and set aside until thick and sticky). But if it’s also acting as a leavening agent along with binding then add in additional ¼ tsp baking powder.

To act as a leavening agent
Beaten eggs are a leavening agent as they incorporate air into the batter, which will expand in the oven and cause the cake to rise. Some cakes use beaten eggs as their only source of leavening.

Examples:
Cookies, Cakes, Cupcakes, Meringues, Soufflés etc

Substitutes:
Baking powder (plus extra oil), Vinegar, Baking soda (plus lemon juice/orange juice), Buttermilk (with Baking Soda) all play an equivalent role as leavening

To provide moisture and texture (Emulsify).
Eggs have emulsification properties, which in plain English would mean something that binds things together which naturally won't come together or mix. Like say- Oil and Vinegar. Egg yolks allow fats to stay dispersed in water and water to stay dispersed in fats which in turn promote thickening and product stability. So this property gives the baked goodies that smooth and creamy texture along with providing it moisture.

Examples:
Mayonnaise (emulsifier) and, Cookies, Muffins, Quick Breads (to provide moisture)

Substitutes:
Turns out that there is lecithin in Tofu as well which acts as an emulsifier when oil the oil is added slowly, which is perhaps why you will see that Tofu is used to make Vegan Mayonnaise. For additional moisture, I tend to use fruit or fruit puree like banana, applesauce, pumpkin puree, apple butter etc

To provide color
The carotenoids present in the eggs are responsible for the color it provides to foods. This is found in Egg yolks (the yellow part), that give crusts that beautiful golden brown color.

Examples:
Scrambled Eggs, fried cutlets, egg wash etc.

Substitutes:
As for cases like scrambled egg, I tend to replace it with Tofu with a pinch of Turmeric. The softness of tofu mimics the creaminess of eggs and the turmeric gives it that beautiful golden color to ape the yellow of eggs. Tofu, as we all know, is a protein powerhouse too.

Other Uses
1. When it comes to using it for frying, I use a mix of all purpose flour + water to form a paste and then apply that as a coating instead of beaten eggs. Instead of all purpose flour, you can also use Chickpea Flour (Besan) to do the same. Other options that I have heard of is using Mayo, Milk, Italian Dressing, Buttermilk etc
2. For Glazing on pies, breads etc, I use Milk/Cream. It provides an even reddish brown color. Using some melted butter to glace also works fairly well. I have heard of Soy milk and melted margarine being used although I have personally not tried this myself.

For extra Volume and Lightness
Egg Whites when beaten provide extra volume and air thereby giving lightness to the finished product.

Examples:
Cakes, Pancakes etc

Substitutes:
One way to do this, that works for me personally, is creaming the butter and sugar in the beginning of the cake making process for a long time until it becomes very light and fluffy. Also, by simply folding the ingredients instead of beating, helps to provide lightness to the baked goods. This is because, both the creaming and folding process helps to incorporate more air into the batter. Of course this is not applicable in case of Angel Cakes/ Sponge cakes where eggs form the basis of such cakes.

All said and done, keep in mind that if a recipe for baked goods calls for three or more eggs per batch (with a typical "batch" consisting of 36 cookies, one pan of brownies, one loaf of bread, or one cake), then, more often than not, egg substitutes do not work. Baked Goods like Pound cakes, sponge cakes, angel food cakes, and other popular desserts with relatively high egg content do not turn out well in egg-free cooking. In these situations, it is usually best to make something else.