Peter wasn't always named Peter. He used to be Simon. But somewhere along the way, Jesus saw something in Simon that prompted Him to give Simon a new name: Peter, the Rock. Even though Peter didn't always act like a rock (i.e. Good Friday), the name stuck. Years later, as "The Rock" wrote his first letter to the Christian believers scattered around the world, he reflected on how his Lord and Master is like a Stone Himself!
Christ is a "living stone," Peter says (vs. 4), and we can imitate Him and become like "living stones" (vs. 5). As a collection of living stones, Peter reveals that the purpose of the Church is to be built into a spiritual house, or temple, for God.
If His followers are stones, then Christ is the cornerstone, the One in whom life holds everything together. But to those outside the faith, Christ is rejected, a discarded stone that becomes a stumbling block to them.
The question for us is: which will Christ be to us? A cornerstone or a stumbling block?