The book of Zechariah mentions a fast that takes place in the tenth month.  Traditionally this is seen as a commemoration of the destruction of Jerusalem, which began in the year 588 BCE:
The word of the LORD came to me in the ninth year, in the tenth month, on the tenth of the month, saying, “Son of man, write the name of the day, this very day. The king of Babylon has laid siege to Jerusalem this very day.” 
The siege was headed up by the notorious Babylonian King Nebuchadnezzar, and eventually led to the total destruction of Jerusalem and the Temple.  Therefore, a fast takes place every year in the tenth month (Tevet) on the tenth day.
This fast lasts from sun-up to nightfall and is a fast from food and water. Because such high significance was placed on this day, it is the only fast that can fall on Erev Shabbat (Friday) which means that one enters Shabbat while fasting.  In the morning, special prayers are recited called Selichot, and the Torah portion containing the incident of the golden calf is read aloud.  Some communities also read Isaiah 55:6 through 56:7 in the afternoon. In Israel those who died at unknown dates in the Holocaust are remembered on this day.
The ultimate purpose of this day can be summed up in the words of Maimonides:
The essential significance of the fast of the Tenth of Tevet, as well as that of the other fast days, is not primarily the grief and mourning which they evoke. Their aim is rather to awaken the hearts towards repentance; to recall to us, both the evil deeds of our fathers, and our own evil deeds, which caused anguish to befall both them and us and thereby to cause us to return towards the good. 
He sees that the true intention of this day is repentance and ties this in with the verse in the Torah, “If they confess their iniquity and the iniquity of their forefathers.”  The Fast of Tevet becomes a perfect day to remember Yeshua’s words, “Repent, for the kingdom of heaven is at hand.” 
- Zechariah 8:19.
- Ezekiel 24:1-2.
- 2 Kings 25:1-3; Jeremiah 52:4-6.
- This was due to God’s words in Ezekiel 24:2 “this very day”.
- Exodus 32:11-14, 34:1-10.
- Rambam, Mishneh Torah, Hilchot Ta’anit 5. Translation from https://www.ou.org/holidays/asarah-btevet/asara_btevet/.
- Leviticus 26:40.
- Matthew 4:17.
Source: First Fruits of Zion