Vegans remove all products that involve animals (including insects) -- for example milk and eggs -- from their lives, usually for ethical and enviromental reasons. There is a bit of variety in the vegan community; Europeans and Americans not only pronounce it differently, European vegans also generally won't eat honey, while American ones generally will.
Most people calling themselves "vegetarians" today do not mean that they are vegans... they mean that they are "ovo lacto vegetarians", who eat milk and eggs.
A more strict version of veganism is the macrobiotic diet.