Adhering to a vegan diet or avoiding eggs for allergy reasons doesn't mean you have to eliminate birthday cakes, chocolate-chip cookies and other baked goods from your diet. Bake cakes and cookies without eggs in a number of ways. If the texture and flavor are correct, the final result will be on par with the regular version of the dish.
In baked goods, eggs are used to bind the dish together and as a leavening agent. Whipping egg whites makes cakes and other baked goods even more light and airy. Any substitute that replicates these properties will make a fine replacement for eggs, so long as it doesn't impart its own flavor or alter the texture. Some egg substitutes such as bananas or applesauce, which help bind baked goods, will alter the flavor slightly, so make sure you use a substitute with a negligible flavor or one that will enhance your final product.
You can purchase egg replacer powders such as Ener-G Foods Egg Replacer at health food stores and online. These products are made of potato starch, arrowroot powder or tapioca starch, all which mimic the binding effects of eggs in baked goods). They also include leavening agents such as calcium carbonate, which makes them a wise choice for recipes in which egg's leavening properties play a major role, such as in meringues. You can make your own egg replacement powder using a combination of arrowroot powder or tapioca starch for binding, along with baking powder for leavening.
To replace one egg in a recipe for a cake or cookies, mash up half a large banana, use 1/4 cup applesauce or blend 1/4 cup of silken tofu into the liquids of the recipe to help bind it. You can also whisk together until foamy 1 tsp. baking powder, 1 1/2 tablespoons warm water and 1 1/2 tbsp. oil to substitute for an egg. The baking powder helps create a lighter texture; you can add 1/2 tsp. of baking powder to applesauce, tofu or banana substitutes to fluff up their texture, too. Another option is flax seeds; 1 tbsp. of ground flax seeds and 3 tbsp. water is equivalent to one egg in baking. However, since flax seeds have a strong taste, make sure to use them only in baked goods where you don't mind the taste of flax, or where it might enhance the final taste, such as in muffins. For any substitution, a rule of thumb to keep in mind is that if the baked good needs to rise, add baking powder.
Another option for an egg substitute that binds is unflavored gelatin. Use it at a ratio of 1 tsp. gelatin to 1 tbsp. warm water. This works well in cookies or cakes; again, remember to add baking powder if a leavening agent is required.
You can also use yogurt to fulfill the binding role of an egg in a baked good recipe. Use 1/4 cup yogurt to replace an egg. You can use plant-based yogurts, such as soy, rice and coconut yogurts, as well as dairy yogurt. Flavored yogurts could be used to add flavor to a recipe if you think that the final combined taste will be a positive experience.
Remember that not all cakes and cookies rely on eggs. You can find a huge variety of egg-less, no-bake cakes and cookies that don't depend on the rising or binding properties of eggs. In these recipes, you won't need to substitute for anything at all.