Islamic New Year is on the first day of the month of Muharram, which is the first month of the Islamic calendar. Muharram is one of four holy months in Islam. During this month all fighting is prohibited, but the celebrations are more cultural than religious. It commemorates Muhammad (pbuh) leaving Mecca and crossing the desert to Medina where he was more free to worship.
Like New Year in other cultures, it is a day for reflecting on the past and resolving to be better in the coming year.