I guess many saints with that name.
To each day some names are tied. Your name day is the day which is tied to that particular name.
I think it's a Christian thing originally. A day of the year is the feast day for a particular saint. For example June 29 is the feast day of Saint Peter and Paul apostles. People named Peter or Paul can celebrate their name day on June 29. Peter for example has many more days, probably because lots of other saints were called Peter, each having their feasts on different days. For example Peter the Exorcist (martyr saint) feast day is June 2, so people named Peter can celebrate their name day on June 2 as well if they wish.
Since then other names of not Christian origin also got their day too. For example Attila has name day on January 7, and October 5.
Furthermore I believe back then it was customary (but not mandatory) to name you after the saint which had its feast on the day you were born. Sometimes people also did not know on which day they were born, so their name day were more important.
Or some shit like that, I'm sure Bernd will correct me.