The word "church" can make a modern reader picture a building, denomination, or later institution. The Greek word ekklesia did not originally mean a church building.
It referred to an assembly or gathered community, and Matthew uses it in Jesus' promise, "I will build my church."
Ekklesia was an ordinary Greek word for an assembly and was also used in the Greek Old Testament for gatherings of God's people. Its New Testament use therefore could draw on both common Greek usage and Israel's scriptural vocabulary.
In Matthew 16 the image is not architectural in the sense of constructing a religious building. Jesus speaks of forming a people or assembly that belongs to him.
The linguistic background shifts attention from a physical structure to a community. Jesus' promise concerns people gathered under his authority.
That is especially significant in a passage concerned with his identity, the rock, the gates of Hades, and the keys of the kingdom. The "church" is first of all the community Jesus himself claims to build.