Eid al-Adha
Eid Al-Adha (عيد الأضحى, "the festival of sacrifice") is the biggest Islamic holiday, in which Muslims kill animals to celebrate a religiously motivated murder attempt by Abraham against his son.
Story of Abraham in the Quran
Religiously motivated murder attempt by Abraham against his son:
My Lord, grant me [a child] from among the righteous."
So We gave him good tidings of a forbearing boy.
And when he reached with him [the age of] exertion, he said, "O my son, indeed I have seen in a dream (في المنام) that I [must] sacrifice you, so see what you think." He said, "O my father, do as you are commanded. You will find me, if Allah wills, of the steadfast."
And when they had both submitted and he put him down upon his forehead
It literally says that it was "in a dream" (في المنام). He tried to kill his son based on a dream and Muslims celebrate this incident.
Hadith
Kill animals
You have to kill animals:
Narrated Al-Bara':
I heard the Prophet (p.b.u.h) delivering a Khutba saying, "The first thing to be done on this day (first day of `Id ul Adha) is to pray; and after returning from the prayer we slaughter our sacrifices (in the name of Allah) and whoever does so, he acted according to our Sunna (traditions)."
Have some music
Muhammad forbids musical instruments and singing girls and said that those who indulge in this will be turned into monkeys and pigs:
It was narrated from Abu Malik Ash’ari that the Messenger of Allah (ﷺ) said: “People among my nation will drink wine, calling it by another name, and musical instruments will be played for them and singing girls (will sing for them). Allah will cause the earth to swallow them up, and will turn them into monkeys and pigs.”
But when it comes to celebrating Abraham's murder attempt, music is ok:
Narrated Aisha:
Abu Bakr came to my house while two small Ansari girls were singing beside me the stories of the Ansar concerning the Day of Buath. And they were not singers. Abu Bakr said protestingly, "Musical instruments of Satan in the house of Allah's Messenger (ﷺ) !" It happened on the `Id day and Allah's Messenger (ﷺ) said, "O Abu Bakr! There is an `Id for every nation and this is our `Id."
"`Id" = "Eid" (the first letter ع has no corresponding letter in English alphabet, so it is transliterated in many ways).
And kill infidels
During Eid al-Adha Muhammad used to give sermons (khutbas) and then send Muslims on military expeditions:
Narrated Abu Sa`id Al-Khudri:
The Prophet (ﷺ) used to proceed to the Musalla on the days of Id-ul-Fitr and Id-ul-Adha; the first thing to begin with was the prayer and after that he would stand in front of the people and the people would keep sitting in their rows. Then he would preach to them, advise them and give them orders, (i.e. Khutba). And after that if he wished to send an army for an expedition, he would do so; or if he wanted to give and order, he would do so, and then depart. The people followed this tradition till I went out with Marwan, the Governor of Medina, for the prayer of Id-ul-Adha or Id-ul-Fitr. When we reached the Musalla, there was a pulpit made by Kathir bin As-Salt. Marwan wanted to get up on that pulpit before the prayer. I got hold of his clothes but he pulled them and ascended the pulpit and delivered the Khutba before the prayer. I said to him, "By Allah, you have changed (the Prophet's tradition)." He replied, "O Abu Sa`id! Gone is that which you know." I said, "By Allah! What I know is better than what I do not know." Marwan said, "People do not sit to listen to our Khutba after the prayer, so I delivered the Khutba before the prayer."