Muharram is one of the holy months. The month of Muharram marks the beginning of the Hijra year and the first of the four months in which war is forbidden. Muharram 10 is of great historical significance.
The world-renowned scholar Imam Ghassali (may Allah be pleased with him) narrated: Some of them are: On this day Adam (A) was created and entered Paradise,  Allah created the Throne and Kurs, created the heavens and the earth and the stars, and created Paradise. On the same day that Khalilullahi Ibrahim (a) was born, he escaped from the fiery furnace of Namrud, and Moses (a) and his companions were freed from the persecution of the rebellious Pharaoh and his followers, and drowned Pharaoh and his companions in the sea. The birth of Prophet Jesus (a), his ascension to the heavens, the exaltation of Prophet Idris (a), the anchoring of Noah's ark on Mount Judi, the royal status of Prophet Solomon (a) and the escaping of Prophet Jonah (a) out of the sea. It was on that day that Prophet Yusuf (a) was released from the abyss and Prophet Ayyub (a) was cured. It was raining for the first time on the tenth of Muharram.
  (Mukashafatul Qulub-422)

Prophet Moses (peace be upon him) and later the Jews fasted on the day of Ashura in gratitude for the consolation given to Prophet Moses (peace be upon him) through the destruction of Pharaoh. The Jews honored the day and celebrated it as a holy day. This is stated in the hadith of Bukhari and Muslim.

Ibn 'Abbas (may Allah be pleased with him) narrated that when the Prophet (peace and blessings of Allah be upon him) came to Madinah and saw the Jews fasting there on the day of Ashura, the Prophet (peace and blessings of Allah be upon him) asked them what this fast was. They replied: '' This is a great day. It was on that day that Allah saved Moses (peace be upon him) and his followers, the children of Israel, and destroyed their enemy Pharaoh and his companions by drowning them. So Prophet Moses (peace be upon him) fasted to show his gratitude towards Allah. So we fast on that day too". On hearing this, the Prophet (peace and blessings of Allaah be upon him) said: So the Prophet (peace and blessings of Allaah be upon him) fasted in Ashura and commanded others to fast.
Here is one thing we need to understand in particular. Many people doubt whether the Prophet (peace and blessings of Allaah be upon him) fasted in imitation of the Jews (taqleed). That's not right. The reality is that it was for some other reason. Imam al-Bukhari (may Allah be pleased with him) and other Muslims (may Allah be pleased with him) narrated from Aisha (R) (may Allah be pleased with her) that the Quraish used to fast on the day of Ashura before Islam - since the Dark Age - and that the Prophet (peace and blessings of Allah be upon him) fasted on that day in Makkah.
The Quraish used to dress the Kaaba (changing the ornamental attire of Kaába) on the day of Ashura. Ramadan fasting was made compulsory in the month of Sha'ban in the second year of the Hijra. On that occasion, the Prophet (peace and blessings of Allaah be upon him) said: If anyone is fasting in Ashura, let him do so ...
If not, then so be it. "
  (Bukhari)

Companion women used to instruct the small children to fast and made them familiar to practice this fast. Imam Muslim quoted a Companion woman from Dubayyiá (ra) as saying that when children asked for food, they would give them toys and divert their attention to complete the fast. The glory of Ashura and the virtue and necessity of fasting on that day are evident from this ...

Ashura fasting is a powerful Sunnah. The sins of the past year are forgiven through it. The Prophet (peace and blessings of Allaah be upon him) said: "I think Allaah will forgive the evil of the year before this fast on the next  Ashurah fasting day."
  (Muslim)

The Prophet (peace and blessings of Allaah be upon him) said that whoever extends the expense of food for whom he has the obligation to look after them, Allaah will extend it for him throughout the year. Imam Bayhaqi (may Allah be pleased with him) and others quoted this from prominent Companions such as Abu Hurairah (may Allah be pleased with him), Jabir (may Allah be pleased with him), Ibn Masood (may Allah be pleased with him) and Abu Sayyid al-Khudri (may Allah be pleased with him). Some people who do not have their own identity have accused this Hadith of not having authentity. That's not right. Al-Hafiz al-Iraqi, an eminent scholar of hadith commentary quoted from Abu Hurairah (may Allah be pleased with him), said: Some of them have been strengthened by al-Hafiz Muhammad bin Nasir al-Baghdadi (ra). In addition, there is a series of hadiths of Jabir (may Allah be pleased with him) that have the conditions of Imam Muslim. Ibn 'Abd al-Bart (may Allah be pleased with him) mentioned this in his book Al-Istidkar.