As some basic Jewish education, what caused the destruction of the First and Second Temples?
There's the "easy" response:
The First was destroyed by the Babylonians.
The Second by the Romans.
However, the Sages in the Talmud have an additional explanation, in order to show how these nations could be allowed to destroy the place where Hashem resided.
Here is the short answer from Rabbi Eliyahu KiTov in the Book of Our Heritage:
"Why was the first Beis HaMikdash destroyed? Because of three sins that were prevalent: idolatry, forbidden relationships, and murder. During the second Beis HaMikdash, the people learned Torah, fulfilled the mitzvos, and engaged in acts of kindness - why then was it destroyed? Because baseless hatred was prevalent. This teaches us that baseless hatred is equivalent to the three cardinal sins of idolatry, forbidden relationships, and murder."
Going into slightly more detail in a different place, Rabbi KiTov says:
"The sages taught that the first Beis HaMikdash was destroyed because of three sins which Israel committed in that generation: idolatry, forbidden relationships, and murder. There was not a place in the Land of Israel where they did not worship idols. Only when seven successive high courts had worshipped idols; when the people had desecrated Shabbos, stopped children from learning Torah, ceased the recital of the Shema in the morning and at night, and no longer felt shame in front of each another [sic], was the Land destroyed.
"The second Beis HaMikdash was destroyed because of baseless hatred among the people; because they shamed scholars; because they didn't differentiate between the great and the small; because they lacked men of strong faith; and because they based their judgments on a strict interpretation of Torah law, and would not make concessions to one another."
To me (and I'm surprised I haven't seen this elsewhere), this says that the violation of the halacha between man and Hashem destroyed the first, but the violation of the interpersonal halacha destroyed the Second. It seems counter-intuitive, but murder and forbidden relationships are crimes against Hashem, according to my understanding of the Sages. These crimes deny his kingship by denying the value of the human life.
Why do I think this is important? You must observe BOTH the ritual and ethical halacha. One is not more important than the other. They are two halves of a whole. When either was ignored, we lost the Temple and the Shechinah's presence among us.
If you want to extrapolate further...the violation of of the laws between man and Hashem took the Temple away from us for approximately 70 years. However, our sinas chinam has caused the Beis HaMikdash to be taken away from us for two thousand years! We even agree that many of the same problems with sinas chinam still exist today.
Perhaps the difference lies in how the two groups of sins are forgiven. As we discuss around the High Holydays, we need only seek forgiveness from Hashem for our sins against Him. However, Hashem does not unilaterally forgive us for our sins against other people unless we seek to make restitution and forgiveness from the person we have sinned against. Because we have refused to make restitution, seek forgiveness, and prevent future sinas chinam, we have no Temple today.
So this Tisha B'Av, begin seeking forgiveness from those you've harmed rather than waiting until Yom Kippur is breathing down your neck.