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Eggs (and how to replace them) always seem to be the big hang-up or confusing part of vegan baking. The good news is, you don't need eggs to have terrific baked goods, they can easy be replaced and substituted out. (That also means you can go hog wild with uncooked batter and not worry about getting sick!)
I've put together this cheat sheet -- a PDF of egg substitutes, when to use them, when not to -- happy baking!
(The picture is not the complete chart--just a glimpse of it.)
Note: Since I don't cook or bake with oil, I haven't tried some of these substitutes out myself -- I just heard about them or read about them. My go-to substitute is bananas or applesauce.