I am a vegan baker and don't use eggs. Eggs are not what makes a cake rise, so I think this is a two part question.

The main job of eggs in baking is binding. The most popular vegan substitute for eggs in baking is ground flax. For every egg called for in a recipe, whisk or blend one tablespoon ground flax (you can choose brown or yellow, I prefer yellow for more neutral flavor and color) with three tablespoons warm water until thickened to the consistency of an egg.

Other egg substitutes are, as mentioned by @nutcakes, fruit or vegetable based. Mashed banana, cooked pumpkin or potato puree, applesauce and apricot sauce are all great as binders, and also add sweetness and richness to the flavor. You can even decrease the fat in your recipe by a few tsp when you use them.

Some people use silken tofu as an egg substitute by I don't bake with soy. But I have tried a tofu chocolate cake that was amazing. It is a good sub for custard-type egg dishes like quiche or pudding.

In terms of leavening, the most popular way to make an egg-free cake rise is a combination of baking soda/baking powder. If you add a tablespoon of vinegar to the liquid ingredients, it will react with the baking soda and cause an even more rise. But be careful what recipe you use it in. I have noticed that vinegar can react negatively in combination with alcoholic ingredients or flax.

Speaking of alcohol, beer is another great way to add lightness to batter. Because it is carbonated it adds air. I use lagers in pumpkin cake and apple muffins, and dark beer in chocolate cake. I love the yeast-y flavor beer gives to baked goods.

Baking powder!

I have a friend who is allergic to eggs. I once made her a loaf of banana bread, where the original recipe called for an egg. I omitted the egg and added an additional 1/4 cup of mashed banana or fruit puree like applesauce. It was good. Recently she bought a product called Ener-g Egg Replacer. She used it according to package instructions, with a brownie mix and it came out rock hard and all the oil squeezed out of it into the pan. Completely unedible. The company does say it works better with homemade products. Then I looked up on the company and there was suggestions for using the ener-g with brownie mix that were completely different proportions and that one worked well and she was happy. You might check out that product.

She makes this cake from a book called Bakin' without Eggs, you might check out this book:

Beth's Chocolate Applesauce Cake

1-1/2 c flour
3/4 c sugar
1/4 c cocoa powder
1 tsp baking soda
1 tsp vanilla extract
1 c applesauce
1/2 c club soda
1/3 c vegetable oil
1Tbsp vinegar

Preheat oven to 350F. Butter an 8x8-inch pan. Mix dry ingred in large bowl. Add remaining ingredients, stir until combined. Pour into pan, bake for 30-35 min until golden. Cool 10 minutes on a rack. Invert onto plate.

Frosting

1/2 cup water
1-1/2 tsp instant coffee granules
1/2 cup firmly packed brown sugar
2-1/4 Tbs butter or margarine
2 oz unsweetened chocolate
2 c powdered sugar
2 tsp vanilla
1/2 cup almonds, chopped

In a saucepan on medium-high heat, boil water and add coffee, brown sugar, buter and chocolate until melted. Remove from heat and cool until worm. Beat in powdered sugar and vanilla; add almonds. When cooled, spread over the cake.

There are lots of great cupcake recipes (that can be baked in a 9-inch round pan instead of a muffin tin) in Isa Chandra Moskowitz's Vegan Cupcakes Take Over the World. An adaptation of their chocolate recipe is here: http://www.nytimes.com/2009/03/08/magazine/08food-t-001.html?scp=6&sq=vegan%20cupcakes&st=cse
The leavening in vegan cakes is usually a combination of baking powder and baking soda.

Hi. My local coop, Weaver Street Market (Carrboro, Chapel Hill and Hillsborough NC) produces this vegan chocolate cake. It's good, although I find it a bit dry. Maybe cut back a wee bit on flour?

http://www.weaverstreetmarket.coop/index.php?option=com_content&task=view&id=259&Itemid=1