Question: "What was/is the Tabernacle of David?"
Answer: The Hebrew word translated “tabernacle” is ohel which means “a tent (as clearly conspicuous from a distance): a covering, (dwelling) (place), home, tabernacle, tent.” There are three main references to the Tabernacle (or Tent) of David: Isaiah 16:5, Amos 9:11, and Acts 15:16, in which the apostle James repeats the passage from Amos. The reference in Isaiah 16:5 is not really about the tabernacle of David itself, but rather to the One from the line of David who would someday sit on the throne and rule over all. This is referring to Jesus.
That leaves two references to the tabernacle of David. In Acts 15:16, while speaking to the Jews, James uses Amos 9:11 to give credence to the recent conversion of the Gentiles in the early church. Many Jews were objecting to this because there was uncertainty as to how the Gentiles were to now keep the Law of Moses. The essential argument from Peter's earlier experience with Cornelius, a Gentile, was that God was also calling Gentiles to Himself. The apostles were not to put on them a burden that no one could ever keep (i.e. the Law of Moses).
From James' words alone it is clear that God's promise through the prophet Amos—that He would "build again the tabernacle of David"—was related to what He was just then beginning to do, namely, visiting the Gentiles to take out from among them a people for His Name. After rehearsing what Simon Peter had just told them—that God had chosen that apostle as the instrument whereby He, for the first time, opened the way of salvation to the Gentiles—James plainly declared that this (God's visitation of the Gentiles to take out of them a people for His Name) agreed with the words of the prophets (in general), and those of Amos in particular. The “tabernacle” referred to here, then, is the house of God open to all, both Jew and Gentile, who seek Him in order to worship in truth.
Amos 9:11 says: "In that day will I raise up again the tabernacle of David, that is fallen." There seems to be reference here to a restoration of the Jewish nation to spiritual life in the end time. There might also exist, during that end time, or into the 1,000 year reign of Christ, a tabernacle like the one during David's day. During David's time the tabernacle (or tent) housed the Ark of the Covenant and was a precursor to the temple that Solomon would build. It was a rectangular house of worship, made with elaborate design. Its presence and functionality, with priests, was a sign of God's favor and presence. When Israel fell away from following the commandments of the Old Covenant, the temple was desecrated and needed to eventually be rebuilt, as described in the book of Ezra.