EID UL FITR

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

Ibn Abidayn said: “The Eid days are thus named because Allah renews His Bounties in them; and He distributes His blessings to His worshipers. From these are Fitr (eating), after food had been prohibited and Sadaqaat al-Fitr (the charity of breaking the fast) to the needy. And on Eid al-Adha, the completion of Hajj with the final Tawaaf of visiting; the meat from sacrifices and many other acts. Also because the custom in Eid is joy, happiness, cheerfulness and gladness.” [Hashiyah Ibn Abidayn (2/165)] ^[1]

Hadith

Anas (R) said: “The Prophet Muhammad (May Allah honor Him and grant Him peace) came to Madeenah and the people of Madeenah had in Jahiliyah two days of play and amusement (they are the day of Niarooz (New Years Day) and the day of Maharajaan (Final Day of the Year). So, The Prophet Muhammad (May Allah honor Him and grant Him peace) said: "I came to you and you had in Jahiliyah two days of play and amusement. Indeed, Allah has replaced them for you by that which is better than them: The day of Nahr (slaughtering) and the day of Fitr (breaking fast)." Musnad Ahmad Vol 3: 103, 178 and 235.  ^[2]

Scholars view

Shaykh Ahmad `Abdurrahmaan al-Bannaa said:

"(They are better because,) the day of Sacrifice and that of Fir are legislated by Allaah (SWT), and are His choice for His creatures. They follow the completion of two of the greatest pillars of Islaam, Hajj (pilgrimage) and fasting. On these days, Allaah (SWT) forgives those who performed Hajj and who fasted, and He sheds mercy on all of His obedient creatures.
 

On the other hand, the days of Nayrooz and Mihrajaan were devised by the people of those times, because of good weather or other passing qualities. 

Eid Greetings

 Jubayr Bin Nufayr said:

"When the companions of Allaah's Messenger(S)  met on the day of Eed, they would say to each other, 'Taqabbala 'llaahu minnaa wa mink (may Allaah accept from us and you.)' " Al-Mahaamiliyyaat with a hasan chain according to Ibn Hajar; see Fath ul-Baaree 2:446

And Muhammad Bin Ziyaad said:

"I was with Abu Umaamah al-Baahilee (R) and some other companions of the Prophet (S). When they returned from the Eed, they said to each other, 'Taqabbala 'llaahu minnaa wa mink.' " Ibn Qudaamah in al-Mughnee 2:259; Ahmad said that this has a good chain. ^[3]

References

[1] Celebrations in Islam by Muhammad Al Jibaly http://www.qss.org/articles/celebrations/toc.html

[2] http://www.ahya.org/amm/modules.php?name=Sections&op=viewarticle&artid=77

[3] http://www.qss.org/articles/celebrations/4-7.htm