In Greek, Peter’s name is Petros, which means “rock” or “stone.” When Jesus says, on this rock (Greek petra), he is making a play on words between Peter’s name (Petros) and the word for rock (petra).
Peter was the foundational "rock" because he was the first to confess Jesus as the Christ and was leader of the apostles. He was the "founding member" of the Church (see Ephesians 2:20–22; Revelation 21:14).
Some scholars think Jesus was talking about himself when he mentioned this rock. But it seems Jesus was referring to Peter or Peter's confession. However, it is clearly the church of Jesus, not of Peter. Jesus himself will build it. So, later church leaders do not need to get their authority from Peter.
Matthew is the only gospel writer who uses the term "church" (also Matthew 18:17). Here, it means "the community of God’s people" rather than the expanded idea of an organized church.
will not prevail against it: The church will not be defeated by Satan's attacks (see Matthew 11:12; 13:24–30, 36–43).