A vegan (by definition, according to the person who made the term) does not eat any animal products at all (if it comes from an animal, including butter, whey, honey, gelatin etc., a veagn doesn't eat it).
Additionally, vegans do not use other animal products such as leather, wool, shells and so on. The basis of a vegan's philosophy is an ethical opposition to using animlas for our own purposes.
Animals do die in order to provide thier eggs and milkproducts- for example, in farms where eggs are taken, it is common practice for all male chicks to be discarded into large dumpsters and suffocated in plastic bags (alongside hundreds of thier fellow chicks) and for milk to be made, there is the whole veal industry and I don't think I need to explain that one
There's a lot more to it, but I'll leave it at that and if anyone has more questions, I'll gladly answer them.