Beit Hamikdash is the Hebrew phrase for the Temple that was first built by King Solomon. A bayit is a house. Mikdash comes from kadosh, meaning holy. In the Tanach, it is usually referred to simply as the "House" in which God especially dwells, a domestic image of the place where God and the Jewish people met in holiness.
The Beit Hamikdash stood in Jerusalem for 500 years until it was destroyed by the Babylonians in 586 BCE, and then again from 516 BCE, until it was burned by the Romans in 70 CE. During that time it was the locus of prayer and sacrifice and the spiritual centre of the Jewish people.
Since the second Beit Hamikdash was destroyed, Judaism has found ways to remember it, while subtly displacing its role. Judaism could not have continued without shifting focus away from it. One obvious example is the three daily synagogue prayer services, which were established to replace the Temple sacrifices.
In this period following the Ninth of Av, many express their hope for binyan Beit Hamikdash, rebuilding of the Temple. Some may feel ambivalent about this. Animal sacrifices are alien to our religious sensibilities today. Rabbi Abraham Isaac Kook, a great 20th century authority, opined that in the rebuilt third Temple there will only be vegetarian offerings. A very different religio-political reality would be required for the temple to be rebuilt now, given the presence of Muslim holy sites on the Temple Mount. Ultimately, the meaning of the Temple is universal: "For my house shall be called a house of prayer for all peoples" (Isaiah 56:7).