What is the significance of the Fast of the Tenth of Teves?
The Aruch Hashulchan relates that this date marked the "beginning of the end" of Jewish history. With the incarceration of Yirmiyahu and besieging of Y'rushalayim, the state of national exile took hold with full force.
The Avudraham holds that (either since this was such a serious and long-impacting occurrence or) since it is referred to in Y'chezk'el as "etzem hayom" that the fast would be obligatory even if it fell out on shabas.
Rav Nachman Cohen makes the thematic connection between this and the event of two days earlier, both of which evidence catastrophic failure to foresee outcomes of our actions (but we'll leave that for a different question's answer).
These are good answers, but the Tenth of Teves also commemorates the events that took place on the 9th of Teves (death of Ezra) and the 8th of Teves (writing of the Septuagint). Due to the hardship of having three fast days in a row these events are fasted for on the Tenth as well.
The source is strange but I think it is the most in depth and interesting I have seen: Asarah B'teves