Vegan cuisine uses no animal products, such as meat, dairy, or eggs. This is more restrictive than vegetarian cuisine, which allows non-meat animal products. All vegan recipes are therefore vegetarian, though not all vegetarian meals are vegan. See Wikipedia's Veganism article for reasons why people are vegan, the history of veganism, and other vegan-related information.
Vegans should be attentive to their intake of vitamins B (especially B12) and D, calcium, iodine, and fatty acids. The Vegan Society's nutrition pages have more detailed information on this subject. See also Vegan nutrition on Wikipedia.
When preparing non-vegan recipes in accordance with a vegan diet, meat, eggs, honey, and dairy are replaced with a substitute ingredient. Substitute ingredients try to replicate the taste and texture of an ingredient or its other useful properties. Substituting for an ingredient works to greater or lesser degrees, depending upon the recipe, and some ingredients are almost impossible to replace. For more information see Vegan Substitutions.
Many vegans choose to avoid specialty substitute ingredients, as these products are often highly processed and high in sodium and allergen ingredients. Cooking with wholefoods such as unprocessed legumes, grains, nuts, seeds, fruits and vegetables can be a healthier approach to veganism. Many non-vegans will also enjoy these foods. Vegans often find that cooking 'real' foods rather than substitutes result in higher quality, better tasting meals that appeal to vegans and non-vegans alike.