A feast is normally the dies natalis — the day of death, understood as the day of birth into heaven. The custom begins with the martyrs and is visible in the Martyrdom of Polycarp, whose church resolved to gather at his tomb on the anniversary. Only three birthdays in the flesh are kept at all: the Lord's, his Mother's and the Forerunner's. The exceptions are worth knowing, because they explain most of the disagreements between calendars: a death date that collides with a Great Feast is moved, as Chrysostom's was; one that falls in Lent may be moved, as Gregory the Great's was in 1969; and a few days keep a consecration instead, as Ambrose's 7 December does.
The Apostolic Fathers with Justin Martyr and Irenaeus (ANF vol. 1)