The Tenth of Tevet is one of the four public fast days (rabbinic as opposed to Torah level) which are built into the Jewish calendar. It is a fast which, like others, commemorates multiple events.
בח' בטבת נכתבה התורה יונית בימי תלמי המלך והחושך בא לעולם שלשת ימים. בט' בו לא כתבו רבותינו על מה. בי' בו סמך מלך בבל את ידו על ירושלים להחריבה.
On the eight of Tevet the Torah was translated into Greek in the days of King Ptolemy, bringing darkness to the world for three days. What occurred on the ninth was not recorded by our rabbis. On the tenth of Tevet the king of Babylon laid his hand upon Jerusalem to destroy it.
The tenth day of the month is when we mark these previous tragic historical events, because on this day the siege of Jerusalem began at the end of the First Temple period, as we find in Ezekiel:
(א) וַיְהִי֩ דְבַר־יהוה אֵלַ֜י בַּשָּׁנָ֤ה הַתְּשִׁיעִית֙ בַּחֹ֣דֶשׁ הָעֲשִׂירִ֔י בֶּעָשׂ֥וֹר לַחֹ֖דֶשׁ לֵאמֹֽר׃ (ב) בֶּן־אָדָ֗ם כתוב־[כְּתָב־] לְךָ֙ אֶת־שֵׁ֣ם הַיּ֔וֹם אֶת־עֶ֖צֶם הַיּ֣וֹם הַזֶּ֑ה סָמַ֤ךְ מֶֽלֶךְ־בָּבֶל֙ אֶל־יְר֣וּשָׁלִַ֔ם בְּעֶ֖צֶם הַיּ֥וֹם הַזֶּֽה׃
(1) In the ninth year, on the tenth day of the tenth month, the word of the LORD came to me: (2) O mortal, record this date, this exact day; for this very day the king of Babylon has laid siege to Jerusalem.
Megillat Ta'anit does not record what occurred on the ninth of Tevet, though by implication it seems something happened. We find a hint of it in later sources.
(ב)... בשמונה בטבת נכתבה התורה יונית בימי תלמי המלך והיה חושך בעולם שלשה ימים בט' בו לא נודע איזה היא הצרה שאירע בו ...
(2) ...On the eigth (of Tevet) the Torah written in Greek during the days of King Ptolemy and darkness was in the world for three days. On the ninth of it, it was not known who caused the trouble that happened on it...
(יג) (יג) לא נודע - ובסליחות שלנו איתא שמת עזרא הסופר:
it was not known - In our selichot penitential prayers it says that on this day Ezra the Scribe died.
The translation of the Torah into Greek is mentioned in many sources, rabbinic and otherwise. Even in the classic rabbinic sources it is not always a clear tragedy.
(ז) מעשה בה׳ זקנים שכתבו לתלמי המלך את התורה יונית והיה היום קשה לישראל כיום שנעשה העגל שלא היתה התורה יכולה להתרגם כל צרכה:
There were once five wise Jews who wrote the Torah in Greek for King Ptolemy, (the Greco-Egyptian King Ptolemy II, reigned 283 - 246 BCE), that day was as difficult for the people of Israel as the day the Golden Calf was made because the Torah could not be fully translated.
ומשום מעשה דתלמי המלך דתניא מעשה בתלמי המלך שכינס שבעים ושנים זקנים והכניסן בשבעים ושנים בתים ולא גילה להם על מה כינסן ונכנס אצל כל אחד ואחד ואמר להם כתבו לי תורת משה רבכם נתן הקב"ה בלב כל אחד ואחד עצה והסכימו כולן לדעת אחת
The Gemara continues: And this was due to the incident of King Ptolemy, as it is taught in a baraita: There was an incident involving King Ptolemy of Egypt, who assembled seventy-two Elders from the Sages of Israel, and put them into seventy-two separate rooms, and did not reveal to them for what purpose he assembled them, so that they would not coordinate their responses. He entered and approached each and every one, and said to each of them: Write for me a translation of the Torah of Moses your teacher. The Holy One, Blessed be He, placed wisdom in the heart of each and every one, and they all agreed to one common understanding. Not only did they all translate the text correctly, they all introduced the same changes into the translated text.
There is a rich relationship between the translation of the Torah into Greek, the death of Ezra and the siege of Jerusalem. Here is a short podcast entitled 'Language under siege' to help you think about the connection.
One last element to reflect upon. The Tenth of Tevet has a special status of never being pushed off to another date. Take a look at this final source and offer some suggestions as to why.
(לה) וחלוק יש ביניהם שארבעת הצומות הם נדחין לפעמים כשחלו בשבת חוץ מי' בטבת שאינו חל לעולם בשבת אבל הוא חל לפעמים ביום ששי ומתענין בו ביום ואפי' היה חל בשבת לא היו יכולין לדחותו ליום אחר מפני שנא' בו בעצם היום הזה כמו ביום הכפורים ... ושאר הצומות כשחלין בשבת ואפי' ט' באב דוחין אותם לאחר השב'.
(35) There is a distinction between them - the four public fast days are pushed off when they fall out on Shabbat, except for the Tenth of Tevet. It never falls on Shabbat but it does sometimes fall out on Friday, when we do observe the fast. Even if it did fall out on Shabbat, we would not be able to push off this fast to another day because it says regarding it "...this exact day..." (Ezekiel 24:2) just as it says regarding Yom Kippur...and when the other fast days fall out on Shabbat, even Tishba b'Av, we push them off until after Shabbat.