There are substitutes, but eggs have very important effects on the texture of dough and are therefore hard to substitute. You will need to experiment a great deal until you hit on the correct texture, because you'll need to tweak all other ingredients and maybe include new ones, e.g. substitute part of the butter and use cream instead to account for moisture loss from egg substitutes.
Mien's link already includes some substitute ideas, like applesauce and commercial imitations. I can't give you more suggestions, because I always put eggs in my cakes ;). But when tweaking, keep following in mind:
Egg whites provide a resilient mesh which traps small air bubbles, so they make a cake more airy. You may need to adjust the baking powder upwards a bit. If it is a soda recipe, you'll need to raise the acidity too. But the whole effect can't be approximated just with baking powder, because its bubbles are too big. For that, you need better emulsification (covered below).
Egg whites also have a binding quality. This shouldn't need compensation in a cake, as you need a dough without too much binding (as opposed to bread), but if your cake starts falling apart, consider using a flour with higher gluten content (all purpose flour instead of cake flour). Gluten binds too.
If, on the other hand, your cake is too bread-like, then you may need less binding. The egg yolks are a source of fat, which prevents the dough gluten from forming a strong mesh by covering the particles. You will miss this with egg substitutes, especially if they are pectin-based (like applesauce) because pectin enhances binding. Use the lower gluten dough (it it doesn't cause the cake to fall apart) or add more fat.
Then there is the importance of emulsifiers. Egg yolks are one of the best sources of lecithin in the kitchen. An emulsifier helps your wet ingredients bind in an emulsion, instead of staying in two separate phases with the watery phase soaking the flour through and the fatty phase making the whole thing greasy. It also helps the trapping of very tiny air bubbles in the wet part of the dough (this is what happens when you beat eggs with sugar on high speed until the colour lightens) which provides a very fine leavening. Maybe commercial egg imitates contain emulsifiers - check the list - but if they don't, you must buy pure lecithin. It is usually soy derived, but maybe you need to check with the manufacturer that your brand is suitable for vegetarians. Also, consider including cream, the fat in it is already emulsified in water, unlike butter (which is water emulsified in fat). You must calculate precisely and tweak other ingredients to ensure that the total amount of fat and moisture don't change (except if you are compensating for the changes caused by the egg substitute).
If it is a chocolate cake, consider reducing the fat and the flour and adding some cocoa powder (the regular-fat variety). Cocoa has binding and thickening properties, and will help with the more even fat distribution. Melted chocolate tastes good in cakes, but makes them very dense, so don't use it.
The easiest way would be to find a recipe published by someone who has already done all the work. But just from paging through a cookbook, you can't tell if they did it right, or just slapped a spoon of applesauce instead of an egg and called it a vegan cake.
