It depends on what the eggs' purpose is in your recipe. Usually, egg provides a combination of moisture, binding, and a little bit of lift, so your substitute's going to have to provide the same.
If you're just replacing one egg, I've found that oil, water, and a little bit of baking powder works great without altering the taste too much. A lot of oil is probably less healthy than a fresh egg though.
You can use mashed banana and/or peanut butter as a binding agent, but be sure to add water and baking powder to cover the liquid and lift jobs, otherwise you'll end up with something too dry and dense. Obviously this will change the flavor a lot.
I've also heard of apple sauce as another vegan substitute, but I've never had any luck getting that to work.