Very technically speaking, a birthday is the anniversary of one's birth. But it is rarely, if ever, referred to in that fashion. "Anniversary" is a word used to commemorate special occasions, like weddings, first dates, very large purchases ("Happy Anniversary! It was one year ago today that we bought this house together.") , things like that. Interestingly enough, even though we don't usually refer to birth dates as "anniversaries" we do use the word when referring to death dates. Not in a celebratory fashion, of course; some people just tend to keep track and mention something like "It's the fifth anniversary of my dad's death. I remember exactly what I was doing that day when the phone rang and my mom gave me the bad news."