Is it allowed in Islam to celebrate birthdays?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Celebrating the birthday of the Prophet (Sallallahu Alaihi wa Sallam) or anyone else is an innovation; there is no evidence approving it. On the other hand, the Prophet (Sallallahu Alaihi wa Sallam) said: "If somebody tries to introduce into this faith of ours (Islam) something which is not a part of it, is to be rejected (and the person is) condemned" .
Thus, a Muslim should not attend such a celebration except to stop or change it by at least talking to the people and informing them with wisdom and good manners. For more details, please read the Fatwa: 83240.
Allah knows best.