Eid al-Mubahila celebrates the famous event in the year 10 AH when a party of Christians, led by the Bishop of Najran called Abdul Masih (or Abu Harisa), decided to debate with the Prophet (peace be upon him and his progeny) about the nature of Prophet Jesus (peace be upon him).
It came after they received a letter from the Holy Prophet inviting them to be Muslims, and meeting and debating amongst themselves about the contents of the letter and the appropriate response.
On the day they had agreed upon earlier, they refused to do Mubahala since they saw that the Prophet (PBUH) came with closest members of his family, who were his daughter, Fatima al-Zahra (SA), his son-in-law, Imam Ali (AS), his grandsons, Hassan (AS) and Hussein (AS), and thus Christians understood his truthfulness. This way the Prophet (PBUH) became victorious in this event.