Eggs perform different functions depending on what you're cooking. In baking, they are usually used for either leavening or binding purposes. You can't replicate a pavlova (that I know of), but you can, with a little egg replacer know-how, manage to adapt most recipes successfully. Try to stick to recipes that require substituting no more than 2 eggs - things just don't seem to work otherwise!
Allergy-conscious/vegan bakers can use any one of the following methods. All quantities specified equal 1 egg..
Commercial Egg Replacer (eg Orgran No-Egg)
This is a good allround egg replacer staple to keep in the pantry. Good for biscuits, cookies, scones, light cakes, pastry and anything that requires a bit of binding (this includes savoury things such as meatballs and burgers though if you omit altogether the meat still keeps firm). It's made from a combination of tapioca and potato starch powders and is beaten together with water. Follow instructions on the packet but usually is 1 teaspoon of No-Egg with 2 tablespoons of water. Beat vigorously, you want it foamy.
1/4-1/2 cup mashed banana
1/2 cup apple puree (or pear)
Good binder/thickener for fruity cakes like carrot or banana, muffins and brownies - anything where the sweetness of the banana or apple won't affect the taste. Usually requires a little baking powder in the recipe or self-raising flour to aid the rising if this is required.
1/2 pack of silken tofu, whipped/beaten
Can be used for cakes (dense ones rather than light sponges) and brownies or uncooked in things like chocolate mousse and fluffy cake frostings. Though in the latter, sweeteners like honey, sugar or maple/golden syrup are essential to disguise the soy taste.
Many vegan bakers use this and it also happens to be a great source of fibre, essential omega 3 fatty acids and antioxidants. Best to buy them whole and ground them yourself though the ground seeds should be stored in fridge or freezer. Good for baked goods where a bit of graininess doesn't detract from the taste eg. muffins, biscuits, cookies, breads or pancakes.
1 tbsp of ground flaxseeds and 1/4 cup water, whisked very well
DIY egg replacer
When you have none of the above in your kitchen and you're desperate, you can use the following:
1 teaspoon baking powder + 2 tablespoons vegetable oil + 2 tablespoons water mixed together = 1 egg