From the Muharram page on Wikipedia:
Muḥarram is the first month of the Islamic calendar. It is one of the four sacred months of the year. Since the Islamic calendar is a lunar calendar, Muharram moves from year to year when compared with the Gregorian calendar.
The word "Muharram" means "forbidden". It is held to be the second holiest month, following Ramadan. Some Muslims fast during these days. The tenth day of Muharram is the Day of Ashura, which to Shia Muslims is part of the Mourning of Muharram.
Sunni Muslims fast during this day, because it is recorded in the hadith that Musa (Moses) and his people obtained a victory over the Egyptian Pharaoh on the 10th day of Muharram; accordingly Muhammad asked Muslims to fast on this day that is Ashura and on a day before that is 9th (called Tasu'a).
Shia Muslims during Muharram do different things and with different intentions. They observe and respect Muharram as the month that martyred Hussein ibn Ali, the grandson of Muhammad and son of Ali, in the Battle of Karbala. They mourn for Hussein ibn Ali and refrain from all joyous events. Unlike Sunni Muslims, Shiites do not fast in this month, especially on the 9th and 10th days of Muharram. In addition there is an important Ziyarat book, the Ziyarat Ashura about Hussein ibn Ali. In the Shia sect it is popular to read this ziyarat on the "Day of Ashura", although most of the Shias try to read Ziyarat Ashura every day and they send salutations to Husayn ibn Ali.