There are two main holidays in Islam, Eid Al-Fitr and Eid Al-Adha. The way that holidays are recognized can vary across cultures, as well as across sects of Islam, Sunni and Shia.
Eid ul-Fitr, often abbreviated to Eid, is a Muslim holiday that marks the end of Ramadan, the Islamic holy month of fasting. Eid is an Arabic word meaning "festivity", while Fiṭr means "to break fast"; and so the holiday symbolizes the breaking of the fasting period. It is celebrated after the end of the Islamic month of Ramadan, on the first day of Shawwal.
Eid al-Adha "Festival of Sacrifice" or "Greater Eid" is an important religious holiday celebrated by Muslims worldwide to commemorate the willingness of Abraham (Ibrahim) to sacrifice his son as an act of obedience to God, but instead was able to sacrifice a ram (by God's command)
Ashura is commemorated by Shi'a Muslims on the ninth and tenth days of Muharram on the Islamic Calendar. Ashura is an Arabic word meaning "ten", and according to Sunni schools of thought it is a day of optional fasting.
Laylat al-Qadr is Arabic for "The Night of Power". It falls on one of the last ten days of Ramadan on an odd numbered day. It is considered the holiest night of the year, since it is the night in which the Qur'an was first revealed. It is also considered better than a thousand months [Qur'an 97:1-3]. It is said that if a person performs voluntary worship on this night, that worship is equal to a thousand months or approximately 80 years.