A birthday cake was a cake served to a person on his or her birthday. A birthday cake was often decorated with small candles. Human tradition held that the person with the birthday might make a wish, which would come true if all the candles could be blown out in one breath.
In 2376, Janeway had to search the databanks for hours on end, but eventually found Lieutenant CommanderTuvok's birthday, and replicated him a cake with a candle on it. He claimed that this was a Human tradition, and Vulcans do not "blow out candles." In the end, he complied, claiming that "it was a fire hazard." (VOY: "Fury")