Cooking an eggless cake
Eggs are added to a cake primarily to help it rise, and to a limited extent to improve the moistness.  The lecithin in the yolk also has emulsifying properties.  For cakes, commercial egg replacers such as Ener-G will usually do the job nicely.

There's also a dairy product called QimiQ that's used in all manner of recipes to substitute for eggs, especially when a recipe needs raw eggs.  It will also work well for cakes, although the substitution tends to be a little more involved.  Also, it's probably harder to obtain.

I wouldn't substitute anything else in a cake.  I've heard of people substituting all manner of things in other baked goods like cookies, but with a cake, you really need it to rise and you need the wet ingredients to bind together with the fats, and the only single "common" ingredient that will do that is either egg or a commercial egg substitute.

If you're literally about to bake this, as in you've got all your other ingredients already sitting in mixing bowls, and need to substitute the eggs with common ingredients you already have in your fridge or pantry, then I'm afraid you may have wasted your time (and your other ingredients).  But maybe somebody else will have some suggestions - guess it can't hurt to wait, given the alternative.