The month of Muharram is one of the sacred months in Islam. It is the first month of Islamic calendar. The Islamic or Hijri calendar is a lunar calendar. It contains 12 months that are based on the motion of the moon. Islamic calendar is consistently shorter than a solar year by about 10 days every year, and therefore it shifts with respect to the Gregorian calendar. 
There are twelve months in Islamic Calendar as mentioned in Qur’an: “Verily, the number of months with Allaah is twelve months (in a year), so was it ordained by Allaah on the Day when He created the heavens and the earth; of them four are Sacred (i.e. the 1st, the 7th, the 11th and the 12th months of the Islamic calendar). That is the right religion, so wrong not yourselves therein” Quran Surah Tawbah 9:36
The Twelve months of Islamic Calendar are as follows:
Abu Bakrah (R), reported that the Prophet Muhammad ﷺsaid: "The year is twelve months of which four are sacred, the three consecutive months of Dhu'l-Qa'dah, Dhu'l-Hijjah and Muharram, and Rajab which comes between Jumada and Sha'ban." Sahih Al-Bukhari 3197(NE) [Vol 4:419 ], 4708 and 5550, Sahih Muslim 1679 and Abu dawud 1947
Abu Huraira (Allah be pleased with him) reported that he (the Messenger of Allah ﷺ) was asked as to which prayer was most excellent after the prescribed prayer, and which fast was most excellent after the month of Ramadan. He said: Prayer offered in the middle of the night and the most excellent fast after (fasting) in the month of Ramadan is the fast in Allah's month al-Muharram. Sahih Muslim 1163 and Jami` at-Tirmidhi 438
The Messenger of Allahﷺwas asked about fasting on the day of 'Ashura (10th of Muharram), whereupon be said: It expiates the sins of the preceding year. Sahih Muslim 1162 (Book 6, Hadith 2603) [Same hadith: Prophet was asked about fasting on the day of 'Arafah (9th of Dhu'I-Hijja), whereupon he said: It expiates the sins of the preceding year and the coming year.]
Ibn 'Abbas (may Allah be pleased with him) said: "The Prophetﷺcame to Madinah and saw the Jews fasting on the day of 'Ashura'. He said, 'What is this?' They said, 'This is a righteous day, it is the day when Allah saved the Children of Israel from their enemies, so Musa fasted on this day.' He said, 'We have more right to Musa(Moses) than you,' so he fasted on that day and commanded [the Muslims] to fast on that day." Sahih al-Bukhari, 2004(Vol.3: 222) and SahihMuslim 1130(Book6 hadith 2520)
Narrated `Aisha(R): Allah's Messenger (ﷺ) ordered (the Muslims) to fast on the day of 'Ashura', and when fasting in the month of Ramadan was prescribed, it became optional for one to fast on that day ('Ashura') or not. Sahih al-Bukhari 2001
Narrated Salim's father: The Prophet (ﷺ) said, "Whoever wishes may fast on the day of 'Ashura'." Sahih al-Bukhari 2000 
These months are called sacred for two reasons:
Hence Allaah has forbidden us to commit sins during these months, as He says: “wrong not yourselves therein” Quran Surah Tawbah 9:36
Although committing sins is haram and forbidden during these months and at other times, in these months it is more forbidden.
Al-Sa’di (may Allaah have mercy on him) said (p. 373):
In the phrase “wrong not yourselves therein”, the pronoun may be understood as referring to twelve months. Allaah states that He has made them a measure of time for His slaves, which they may use for worshipping Him, and thank Allaah for His blessings, and they serve the interests of His slaves, so beware of wronging yourselves therein.
The pronoun may also be understood as referring to the four sacred months, and this forbids them to wrong themselves in those months in particular, as well as it being forbidden to do wrong at all times, because it is more forbidden at this time, but it is worse at this time than at others.