A `Eid is any day of gathering. It is derived from `Aada (meaning returned), because people return to it periodically. Some scholars say that it derives from `Aadah (custom or practice) because people are accustomed to celebrating it. Its plural is A`yaad. Ibn ul-`Araabee said: "It is called `Eed because it returns every year with renewed happiness.” Lisaan ul-`Arab 
The Eid prayer is one where the imam attends and leads the people in praying two rak’ahs. ‘Umar (may Allaah be pleased with him) said: “The prayer of al-Fitr is two rak’ahs and the prayer of al-Adha is two rak’ahs, complete and not shortened, on the tongue of your Prophet, and the one who fabricates lies is doomed.” Narrated by al-Nasaa’i, 1420 and Ibn Khuzaymah. Classed as saheeh by al-Albaani in Saheeh al-Nasaa’i. 
One of the rulings of Eid is that the prayer comes before the khutbah, because of the hadeeth of Jaabir ibn ‘Abd-Allaah who said, The Prophet (peace and blessings of Allaah be upon him) came out on the day of al-Fitr and started with the prayer before the khutbah. Narrated by al-Bukhaari, 958; Muslim, 885. 
Shaykh Ibn ‘Uthaymeen said in Majmoo’ al-Fataawa (16/214):
What I think is that the Eid prayer is fard ‘ayn (an individual obligation), and that it is not permissible for men to miss it, rather they have to attend, because the Prophet (peace and blessings of Allaah be upon him) enjoined that. He even commanded the women – including virgins and those who usually stayed in seclusion – to come out to the Eid prayer, and he commanded menstruating women to come out to the Eid prayer, but told them to keep away from the prayer-place itself. This indicates that it is confirmed.
He also said (16/217): What seems more likely to be correct in my view, based on the evidence, is that it is fard ‘ayn (an individual obligation), and that it is obligatory for every male to attend the Eid prayer apart from those who have an excuse. 
It is permissible for Muslims to greet each other on the day of Eid with specific greetings, which have been reported from the Sahabah (radhi allahu anhu). Responding to a question in this regard, Ibn Taymiyyah (rahimahullah) said: “As for the greetings on the day of Eid, then it is by saying to one another when meeting after the prayer: “'TaKabbala Allahu minnaa wa minkum wa Ahaala Allaahu 'alaika' [Majmoo al-Fatawa (24/253)] “May Allah accept it from you and us and may Allah make it return to you, and other things of this nature.”
Haafidh Ibn Hajr said: “We have related in 'Mahaamiliyaat' with a chain of narration, which is hasan, from Jubair bin Nufair, who said: “The Companions of Allah's Messenger (sallallahu alaihi wa-sallam) when they met on the day of Eid, would say to each other: ‘Taqabbala Allahu minna wa minkum' “May Allah accept it from you and us” [Fath al-Baree (2/446)]
This is the greeting with which the Sahabah (radhi allahu anhu) used to greet each other with. Therefore, greeting one another in this manner is a part of the religion and thus one will be rewarded for it by Allah.