In general, birthdays can originate from these sources:
- Manually entered in your calendar
- From your iOS device contacts (visible in a separate equally named category at the bottom of all your calendars).
Facebook will enrich your contacts with birthdays too.
Duplicate birthdays can be caused by:
- A duplicate contact, resulting in a birthday for both
- iCloud duplicating your birthdays.
To reset your sync:
- Go to System Settings
- Go to Contacts (or Mail, Contacts, Calendars in iOS9 and below)
- In the Accounts section select the one containing your Contacts (normally iCloud one)
- Turn OFF Contacts
- Reboot your device
- Turn ON Contact again