When Christianity became strong, the Church abolished birthday parties as a whole but, since they insisted on celebrating Christ's birthday every year from the fourth century on - hence Christmas -, the tradition found its way back into daily life.
It's cool to know some of the things that came from that time: the custom to put candles on top of cakes, for example, came from Greece, where it represented the Godess Artemis and her full moon. The first baloons were made out of dead animals' viscera, and even back then they were used to entertain kids. To give presents to the birthday person is a custom we inherited from Greeks and Egyptians; they were used in the past, however, to depart evil spirits. Common gifts were myrrh and frankincence.