The Fast of the Firstborn is part of the Jewish celebration of Passover. This part of the holiday commemorates the fact that none of the Jewish families in Egypt were affected by the tenth and final plague God visited upon Egypt. In this plague, all of the firstborn male sons from each Egyptian family were killed.
As a way of thanking God for sparing the Jewish firstborn sons during this plague, firstborn sons of Jewish families that celebrate Passover will often participate in the Fast of the Firstborn. This fast takes place the day before Passover begins. Boys who are younger than 13 aren't expected to fast, but their fathers will typically fast in their stead.