The 10th of Muharram is called Ashura (ʿĀshūrāʾ), an important day in Islam.
For Sunni Muslims:
It is recommended to fast on Ashura.
The Prophet Muhammad ﷺ said that fasting on Ashura expiates the sins of the previous year (minor sins). This is reported in Sahih Muslim.
The Prophet ﷺ also encouraged fasting an additional day with it (either the 9th or the 11th of Muharram) to distinguish Muslim practice from that of the Jews.
Common fasting options:
9th and 10th Muharram (best-known practice)
10th and 11th Muharram
9th, 10th, and 11th Muharram (also recommended by many scholars)