About the name "Peter" meaning "rock"

[JSojourner]

I believe the original Greek name was Petras, which is literally translated "rock".  If I am not mistaken, there is a famous natural landmark in Palestine that came to be called "Petra" because it is a giant rock formation.

I could be wrong about all this, though.

I will let my Roman Catholic brothers and sisters comment about the significance of this in terms of Catholic church polity and structure.

And you're partially correct.  Jesus, or God, have been referred to as a "rock".  The "rock" of our salvation and the "rock" in which we hide (referring to the Old Testament story of Moses being hidden in the cleft of the rock as God's glory passed by).  Jesus is specifically called "cornerstone" and also, of particular interest to me, is the repeated statement that he is a "stone of stumbling".

I take this to apply to those who regard Him as a good teacher, a great man, a prophet, a revolutionary or an honorable example to follow...but cannot accept His own claim to be God in the flesh, a sacrifice for the sins of humankind and risen from the dead.  Hence, they stumble over Jesus.