Muharram is the first month of Islamic year. It’s one of the months of Respect among the four in which fighting is not allowed.
Whilst the best and compulsory month for Fasting is Ramadan, to Ahadees of Rasool Allah (saw) says that after Ramadan, the next best month for fasting is Muharram.
Holy Prophet (Saw) said in Sahih Muslim: The best month for fasting after Ramadan is fasting in Allah’s month of Muharram.
In Arabic the word Ashura means 10th and Fasting on this day is even more important according to some ahadees; Holy Prophet Muhammad (Saw) said “It atones for the sins of the preceding year.” (Sahih Muslim)
Regarding Fasting on 9th and 10th Muharram or 10th and 11th Muharram instead of only on 10th Muharram:
When the Prophet (Saw) fasted on the day of Ashura and commanded that it should be observed as a fast, his companions said to him: “Messenger of Allah, it is a day which the Jews and Christians hold in high Esteem.” Thereupon the Messenger of Allah (Saw) said: “Next Year, Allah willing, we would observe fast on the 9th Muharram as well.” But the Messenger of Allah (Saw) died before the advent of the next year.