What is Eid-e-Mubahala and why only Shi'i celebrate it?
There is no such a thing as Eid-e Mubahala promoted by the Prophet (sws) as the part of the religion. There are only two Eids in Islam: Eid al-Fitir and Eid al-Adh'ha, as it is evident from the Sunnah of the Prophet (sws).
Shi'i Muslims believe that the incident of Mubahala proves an extra ordinary religious position for Ali, Fatima, Hassan and Hussain (rta). This is the incident in which the Prophet (sws) is ordered in verse 3:61 to invite the Christians of Najran to a challenge and a test in which each of the two sides would curse the liars to see who will be saved and who will be harmed as the result. Those Shi'i Muslims who celebrate this incident are in fact appreciating and celebrating what (in their view) this verse implies about the above great personalities.
Would you please send me the details of Mubahala and how the Christians of Najran accepted challenge and what was the outcome of this Mubahala. I will appreciate your early reply.
You can read the story of Mubahila from Tafseer of Ibn Kathir on the verse of Mubahila. A translation of this in English can be foudn here:
Note that ibn Katheer has reported the story from Ibn Is'haq. Ibn Is'haq has not included in his report that Ali, Fatima and Hasanain (ra-hum) had been called by the Prophet (sws) after revelation of this verse.
That part of the story can be found in the hadith collection of Muslim, see the last part of the Hadith:
Book 031, Number 5915:
This hadith has been narrated. on the authority of Shu'ba with the same chain of transmitters. Amir b. Sa'd b. Abi Waqqas reported on the authority of his father that Muawiya b. Abi Sufyin appointed Sa'd as the Governor and said: What prevents you from abusing Abu Turab (Hadrat 'Ali), whereupon be said: It is because of three things which I remember Allah's Messenger (sws) having said about him that I would not abuse him and even if I find one of those three things for me, it would be more dear to me than the red camelg. I heard Allah's Messenger (sws) say about 'Ali as he left behind hrin in one of his campaigns (that was Tabuk). 'All said to him: Allah's Messenger, you leave me behind along with women and children. Thereupon Allah's Messenger (sws) said to him: Aren't you satisfied with being unto me what Aaron was unto Moses but with this exception that there is no prophethood after me. And I (also) heard him say on the Day of Khaibar: I would certainly give this standard to a person who loves Allah and his Messenger and Allah and his Messenger love him too. He (the narrator) said: We have been anxiously waiting for it, when he (the Holy Prophet) said: Call 'Ali. He was called and his eyes were inflamed. He applied saliva to his eyes and handed over the standard to him, and Allah gave him victory. (The third occasion is this) when the (following) verse was revealed:" Let us summon our children and your children." Allah's Messenger (sws) called 'Ali, Fitima, Hasan and Husain and said: O Allah, they are my family.