Vegan really isn't very complicated, and really, that's the point. I think a better, less-stigmatized term for the same thing is "plant-based diet." The latter term is also self-descriptive, and it's a good way to think about preparing foods for vegans to eat. If it is made from or derived from animals or animal products, don't use it to prepare your food. This includes eggs, milk (and cheese, by extension) and fish (fish are not plants). There are plenty of great foods you can prepare using only fruits, vegetables, grains, beans, herbs, spices, etc.
Using animal protein as a primary player in your diet tends to leave your body in an overly acidic state and leads to chronic inflammatory conditions.