Strangely the Prophet Muhammed did not know about this fast. This must mean it must be a "minor" fast in the Jewish calendar and not one obligatory in the Torah such as Yom Kippur (Day of Atonement).
Jewish scholars think the fast day of the tenth of Ashuraa is Yom Kippur, however I beg to differ.
The fast day, the tenth of tevet, is known in hebrew as Asarah BeTevet. Compare this with the Arabic, Ashura. The names are very similar and both are fast days. Logic would conclude that they are the same day.
From the Islamic booklet, A Paper on Hadiths of Month of Muharram By Sheikh Abdullah ibn Saleh Al-Fawzan
The Reason behind Ashuraa' FastingThe fast of tenth of Tevet does not however commemorate the deliverance of the people of Israel from the Egyptians. That happened during Spring. "The fast commemorates the siege of Jerusalem by Nebuchadnezzar II of Babylonia—an event that began on that date and ultimately culminated in the destruction of Solomon's Temple" according to Wikipedia. Is it possible the hadith was not entirely accurate on the particular details since its transmission took place some two centuries after the death of the Prophet Muhammed?
On the authority of Ibn Abbas, may Allah be satisfied with them, he said, "The Prophet, may the blessings and peace of Allah be upon him, came to Medinah and found the Jews to be in the habit of fasting Ashuraa'. When asked about such habit, they said, ‘This is the day when Allah granted Moses and the Children of Israel victory over Pharaoh, hence our grateful fasting.’ Thereupon the Prophet, may the blessings and peace of Allah be upon him, said, 'We are worthier of Moses than you,’ hence the command to fast it," reported by Al-Bukhari and Muslim. In another narration by Muslim, "So Moses thankfully fasted it, and we are in likewise," (19).
The above Hadith clarifies the reason behind fasting Ashuraa', which is holding this day in high esteem and as a manner of thanks offering to Allah Almighty as He saved Moses, may the blessings and peace of Allah be upon him, and the Children of Israel on that day brought Pharaoh and his followers to a detrimental end. Hence, Moses fasted the day as a means of thanks-offering, and the Jews followed in his steps. Yet, the followers of Mohammed are much worthier of taking Moses as an example than the Jews. If Moses had so fasted that day, then we are too. Therefore, the Prophet, may the blessings and peace of Allah be upon him, said, "We are worthier of Moses that you," while another narration reads, "I am worthier of Moses than you," which means that we are closer to Moses' stance and example than you (the Jews) are since we have the same origins of law and believe in his Book, while you are on the opposite page because of changes and perversions you made thereto. The Jews are nothing like our Prophet in terms of righteousness and holding tight to the right path, hence fasting the day and commanding Muslims to.
Nevertheless the similarities in the Hebrew and Arabic would have the 10th of Ashuraa has the 10th of Asarah BeTevet. This date falls in December/January according to our calendar. Muharram is the beginning of the year in the Islamic calendar. Says Timeanddate: "Muharram is the month at the beginning of the Islamic year. Some Muslims mark the start of the Islamic year on the first day of Muharram. Others start to prepare for the Day of Ashura on the tenth day of the month. This observance applies to Muslims worldwide, including the United Kingdom." Ramadan falls in the ninth month on the Islamic calendar. This makes it closer to August/September than the current system of floating the festival during any time of the year. The Night of Power must conclude the festival being the night of Rosh Hashanah where the "rams" horn is blown throughout the day. The month of fasting is a time of repentance which leads up to the time of Judgement which happens at Rosh Hashanah.