Answered by Shaykh Faraz Rabbani
The Significance of Eid al-Fitr
Islam has two major holidays, Eid al-Fitr (Post-Fasting Festival) and Eid al-Adha.
The word Eid itself is an Arabic word, whose root connotation is that which comes back, time after time, and rejoicing. Its particular usage in Islam, for the two major holidays, is because these two days are meant to be days of rejoicing. 
The Prophet Muhammad (Allah bless him and give him peace) said, They are days of eating, drinking, and remembrance of God. [Reported by Bukhari in his Sahih, an authoritative collection of the sayings of the Prophet.]
In this same spirit, the Quran mentions that, Jesus, son of Mary, said: O Allah, Lord of us! Send down for us a table spread with food from heaven, that it may be a feast (eid) for us, for the first of us and for the last of us and a sign from You. Give us sustenance, for You are the Best of Sustainers. (Quran, 5: 114)
Eid al-Fitr celebrates the completion of the month of Ramadan, in which Muslims fast and increase their spiritual devotions, and is meant to be a recognition the material and spiritual favors of God to His creation.
On this day, Muslims all over the world thank God for the gift of fasting, in which they avoided food, drink and intercourse from dawn to dusk, out of obedience and servitude. The Prophet Muhammad (Allah bless him and give him peace) said, Whoever fasts the month of Ramadan out of faith, seeking its reward, shall have all their past sins forgiven. [Also reported by Bukhari in his Sahih, and others]
The many lessons in Ramadan are acted upon on this day of festivity, in order that they not be forgotten:
1. Devoting oneself to God: Muslims start the day by showering after dawn on Eid day, then go to the short Eid prayer and sermon that takes place early in the morning.
2. Recognizing ones blessings and thanking God for them: Muslims are encouraged to wear their best clothes, give gifts (especially to children) and celebrate with family, friends, and neighbors.
3. Remembering the plight of the poor and giving in charity: On Eid day, it is especially recommended to give in charity, the best time of which is before going to the mosque or prayer hall in the morning.
Here in Jordan, we gather after the Eid prayer with fellow students, greet each other, and then have a short group dhikr (remembrance of God) that is accompanied by spiritual songs. Before people part, they drink coffee and have local biscuits and sweets.
In this and other ways, Muslims seek to join between worldly and spiritual celebration, for it is said, True rejoicing is not (merely) in wearing new clothes, but in becoming true in ones devotion to God.
As a result, it is encouraged for Muslims to fast another six days after Eid during the month of Shawwal, in order to keep alive the lessons learned during the month of Ramadan, and to become of those devoted to God. It is because of this that the Prophet (Allah bless him and give him peace) said: Whoever fasts of Ramadan then fasts six days in the month of Shawwal shall have the reward of having fasted the whole year. (Sahih Muslim)
The Prophet (Allah bless him & give him peace) said, For every people there is a feast and this is our feast. [Reported by Bukhari in his Sahih]
 Raghib al-Isfahani, Mufradat Alfaz al-Quran, 594 (Damascus: Dar al-Qalam, 1997). This is a classic work on the vocabulary of the Quran.