Yom Ashura or Ashura (عاشوراء ʻĀshūrā’) is the tenth day of Muḥarram, the first month in the Islamic calendar. Ashura marks the climax of the Remembrance of Muharram, and commemorates the death of Husayn ibn Ali, the grandson of the prophet of Islam, Muhammad at the Battle of Karbala on 10 Muharram in the year 61 AH (October 10, 680 CE). Mourning for the incident began almost immediately after the Battle of Karbala.
Fasting the day of Ashura is called by our Imams as (The Fast of Ibn Marjana) according to the narrations. Ibn Marjana was the commander of the army of Yazeed who killed Imam Husain (AS).
It is Makrouh (Disliked) to fast that day. Ibn Marjana and his people wanted to celebrate the day of Ashura to make Muslims forget their crime, that it why they paid lot of money to false narrators to fabricate narrations encouraging people to fast and celebrate that day. We have evidence that narrations encouraging fasting on the day of Ashura are fabricated by enemies of Ahlul Bayt (AS).
The recommended on the day of Ashura is to remember the tragedy of Karbala and weep and cry on it and refrain from eating and drinking till after noon. This is called Faqa.
It is permissible but it is strongly disapproved of (makruh) by the Imams. However, it is recommended to eat or drink simply towards the end of the day such that one is not considered fasting, rather than indulging.
It is good to keep in mind that the custom of fasting on Ashura relates to celebrations by the Umayyads over their victory and also false narrations claiming that the day of Ashura was significant for other reasons (apart from the martyrdom of Imam Husayn) which were attempts to deflect attention away from Imam Husayn.
It is not obligatory but it is recommended. It is considered to be an appropriate form of expression on Ashura, and a means of attracting divine blessings and linking the followers of the Ahl al-Bayt to the Ahl al-Bayt.
Here are some hadith that may be helpful: https://www.al-islam.org/forty-hadith-on-azadari/ahadith-traditions
Sh'is do not accept the hadith saying to fast on Ashura as valid. They are considered fabrications which came about in/after the time of Yazid to promote Ashura as a day of celebration. (Indeed, in some areas, Ashura is still celebrated as a holiday.)
For more information, you can look at resources such as this: https://www.al-islam.org/understanding-karbala-allamah-sayyid-saeed-akhtar-rizvi/appendix-4-fasting-ashura
The first ten days of Muharram are the days of the sufferings of Ahlul Bayt (AS) which started on the arrival in Karbala on the second day of Muharram and ended on the tenth day of Muharram which is called Aashura, when Imam Husain (AS) was martyred along with his family members and companions in Karbala in year 61 After Hijra. These ten days are been observed by Ahlul Bayt (AS) and their followers as days of mourning and grief. Imam Reza (AS) said that his father Imam Musa Al-Kadhim (AS) was always observing these ten days with grief and sadness.
Followers of Ahlul Bayt (AS) observe these ten days with Majaalis (religious gatherings and sermons) remembering the tragedies of Karbala and the aims and teachings of Imam Husain (AS). Millions of believers attend these gatherings all over the world, and thousands of lectures and sermons are delivered to enlighten people and purify their behavior and make them better human beings. Millions of poor people are been fed during these ten days as a practice of generosity which is part of the teachings of Imam Husain (AS).
It is not prohibited. However some ulama says it is mostahab to make day of Ashura, day of sorrow and do not eat so much during the day.