How do we decide which festivals are allowed to be celebrated by Muslims?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

Perhaps the honorable questioner is asking about the way one can judge whether or not a certain occasion is considered to be Islamic and thus Muslims are allowed to celebrate it.

If this is the case, then you should know that festivals ('Eed) were determined by Sharee'ah (Islamic Law) through the narrations reported on the authority of the messenger . Anas narrated that when the Prophet,  arrived to Al-Madeenah, people had two days which they used to celebrate, so he inquired: "What are these two days" People said: 'These are two days we used to celebrate during our pre-Islamic era', Thereupon, he said: "Indeed, Allaah Has replaced them with two days that are better than them; 'Eed Al-Adh-Ha and 'Eed Al-Fitr" [Abu Daawood].

No other days, other than the two mentioned in the narration, were reported as festivals ('Eed) in Islam. Therefore, the fault of those who imitate the disbelievers by celebrating their festivals becomes evident. This also applies to those who innovate new festivals, such as Mother's Day, Birthdays …etc.

To conclude, Sharee'ah is the only source from which one rules whether or not a certain occasion is a lawful festivals to celebrate.

Allaah Knows best.