Why do we do "the passing of the peace"?
The Peace is one of the ancient parts of the Eucharistic liturgy. We pass the Peace for several reasons, many of them biblical.
We pass the Peace because Jesus told us to. In the Sermon on the Mount (Matthew 5:23-24), Jesus said, "So when you are offering your gift at the altar, if you remember that your brother or sister has something against you, your gift there before the altar and go; first be reconciled to your brother or sister, and then come and offer your gift." Passing the Peace began as a way for people in the Christian community to be reconciled to one another before making their offering at the altar. It is for this reason that the Peace always comes before Communion.
We also pass the Peace because at the Last Supper, Jesus said "Peace I leave with you; my peace I give to you" (John 14:27), and later he said, "My command is this: Love each other as I have loved you." (John 15:12). When Jesus himself appeared to the disciples after his resurrection, he greeted them by saying, "peace be with you" (Luke 24:36; John 20:19, 26). We bring these two together: just as Jesus shared his peace with us, so we should share peace with one another.
Sharing peace with one another is an ancient Christian tradition, not only in the words of Jesus, but also in the practices of the Christian community. Paul begins every one of his letters by saying, "Grace to you, and peace" (Romans 1:7; 1 Corinthians 1:3; 2 Corinthians 1:2; Galatians 1:3; Ephesians 1:2; Philippians 1:2; Colossians 1:2; 1 Thessalonians 1:1, etc.). This suggests that in the very early Church, "Grace to you and peace" was the way Christians greeted one another; it was, if you will, their secret handshake.
So the next time you are passing the Peace in a worship liturgy, don't think of it as the intermission, or time-out, when we simply shake hands and say "hello." It is a time when we remember who we are: the reconciled people of God whom Jesus welcomes to his table with words of peace.