How do I delete a birthday from my calendar and contact?

My first thought would be to look up that contact in Contacts.

Once you get them pulled up, look and see if there is a birthday showing for them. If there is, and you don't want it, click Edit in the upper right-hand corner of the screen and then scroll down and delete the birthday.

There is another possibility though...

The birthday could be coming from an outside synced calendar (i.e. Outlook, Hotmail, Yahoo, or gmail). You'd need to find it in the outside calendar and delete it from there so that it won't sync into your phone's calendar.