Papacy
Matthew 16:16-19
Thou art Christ, the Son of the living God... Thou art Peter; and upon this rock I will build my church... I will give to thee the keys of the kingdom of heaven.
Context of the passage
At Caesarea Philippi, after Peter's confession of faith, Jesus renames Simon 'Rock' (Kepha/Petros) and confers on him a unique role in the Church.
Catholic reading
The wordplay in Aramaic (Kepha/Kepha) makes Peter, not merely his faith, the foundation stone under Christ; the keys symbolize delegated authority, the basis for the papacy as Christ's vicar on earth.
Common objection
Some argue the 'rock' refers to Peter's confession of faith, not Peter himself.
Response
The Aramaic underlying text used the same word 'Kepha' for both 'Peter' and 'rock,' a direct identification; moreover, Jesus addresses Peter in the second person throughout ('thou art... I will give to thee'), personally, not abstractly.
The Church Fathers
“On him [Peter] he builds the Church, and to him he gives the command to feed the sheep... a primacy is given to Peter.”