It's easy to replace eggs when baking. Why? Because most recipes that call for eggs don't even need them (however, some recipes really do depend on eggs, such as rich flourless cakes and meringues). In most recipes, an egg just acts as a binder; luckily there other foods that can bind (see my list below).
Here is a little history lesson for you: eggs were only added to boxed cake mixes to make them seem more "wholesome". Originally, boxed cake mixes sold terribly and it was hypothesized that their poor sales were due to the powdered eggs (warning: sexist comments are included in this quote):
"The problem, according to the psychologists, was eggs. Dichter, in particular, believed that powdered eggs, often used in cake mixes, should be left out, so women could add a few fresh eggs into the batter, giving them a sense of creative contribution. He believed...that baking a cake was an act of love on the woman's part; a cake mix that only needed water cheapened that love." (Source: http://www.foodtimeline.org/foodcakes.html).
...hence why boxed cake mixes now call for real eggs to be added.
Some egg replacements include (all equivalent to one egg): 1/2 a mashed banana, 1/4 cup of apple sauce or pumpkin puree, 1 heaping tablespoon ground flax seed (mixed in 1/4 cup water and left to sit for 5-10 minutes), and 2 tablespoons arrowroot or corn starch mixed with 2 tablespoons water.
Why do we want to replace eggs in baking? Well, for those with high cholesterol, eggs can be troublesome as they are very high in bad cholesterol. Also, a lot of people are allergic to eggs. Lastly, not everybody has a carton of eggs lying around when the urge to bake strikes, therefore there is a need for easy egg replacements!