This year Ashura is celebrated/ observed on Sunday October 1st, 2017. Ashura marks the 10th day of the 1st month of Muharram on the Islamic lunar calendar. The day commemorates for Shia Muslims the death of Husayn ibn Ali who was the grandson of Muhammad in the Battle of Karbala of October 10, 680 CE. For Sunni Muslims the day relates to when Moses and his followers fled Egypt and where saved by God separating the Red Sea.
For Shia Muslims this is a day of sadness and morning where some people will make pilgrimages to a shrine (Mashhad al-Husayn) in Karbala Iraq. Some Sunni Muslims use this day to morn the loss of Husayn and other use the day for fasting because they believe God gave victory to Moses on this day by parting the Red Sea.
The date of Ashura always falls on the 10th day of the 1st month of Muharram according to the Islamic calendar. The date is different each year on the Gregorian calendar system because it is based off of the solar cycle vs the lunar cycle for the Islamic calendar.