Muharram Festival refers to the first month of Islamic Lunar calendar. The most important of all dates in this month is the tenth day of Muharram, according to the Muslims. The 10th of Muharram is the day when Prophet Musa gained a victory over Egyptian Pharaoh. The day is usually celebrated with fasting (Ashura fasting), as suggested by Prophet Muhammad. Ibn Abbas narrated that he (Prophet Muhammad) came to Medina and saw the Jews fasting on the tenth day of Muharram. Then he asked to the Jews, "What is this?" They said, "This is a good day. This is the day when Allah saved the children of Israel from their enemy and Musa fasted on this day." Then Muhammad said, "We have more claim over Musa than you." So he fasted on the day and told the people to fast (Sahih Bukhari Book 31 Hadith 222, Book 55 Hadith 609, and Book 58 Hadith 279; Sahih Muslim Book 6 Hadith 2518, 2519, 2520).

Another tradition in Ashura day is held by Shia community (which is often mistakenly regarded as part of Islam). For them, the tenth day of Muharram is the day of mourning. They commemorate it as the day of Karbala, where their Imam, Imam Hussain Ibn Ali was being assassinated and slain mercilessly by Yazid I, the Umayyad caliph. To commemorate this heartbreaking event and the memory of Imam Hussain, Shia community usually adorn black clothes in Ashura day. They gather and arrange massive processions where they cry very loudly and beat their own chest in order to commemorate all the sufferings Imam Hussain has been in. In certain times, some Shia sects also fast for the whole month and flagellate themselves with sticks and rods on this Ashura day. They harm themselves to commemorate the sufferings of Hussain Ibn Ali while fighting against Yazid I oppressive regime, which Imam Hussain found it rather far away from the pure Islamic teachings and values.

Source:
https://travel.earth/muharram-the-festival/
https://en.wikipedia.org/wiki/Ashura