The kiss of peace is an ancient traditional Christian greeting. Also called "Holy kiss", "Brother kiss" among men, or "Sister kiss" among women. Some congregations do not perform inter-gender holy-kissing. It is not the Socialist fraternal kiss.
It was the widespread custom in the ancient western Mediterranean for men to greet each other with a kiss. That was also the custom in ancient Judea and practiced also by Christians.
However, the New Testament's reference to a holy kiss (en philemati hagio) and kiss of love (en philemati agapēs) transformed the character of the act beyond a greeting. Such a kiss is mentioned five times in the New Testament:
The writings of the early church fathers speak of the holy kiss, which they call "a sign of peace", which was already part of the Eucharistic liturgy, occurring after the Lord's Prayer in the Roman Rite and the rites directly derived from it. St. Augustine, for example, speaks of it in one of his Easter Sermons:
Then, after the consecration of the Holy Sacrifice of God, because He wished us also to be His sacrifice, a fact which was made clear when the Holy Sacrifice was first instituted, and because that Sacrifice is a sign of what we are, behold, when the Sacrifice is finished, we say the Lord's Prayer which you have received and recited. After this, the 'Peace be with you’ is said, and the Christians embrace one another with the holy kiss. This is a sign of peace; as the lips indicate, let peace be made in your conscience, that is, when your lips draw near to those of your brother, do not let your heart withdraw from his. Hence, these are great and powerful sacraments.
Augustine's Sermon 227 is just one of several early Christian primary sources, both textual and iconographic (i.e., in works of art) providing clear evidence that the "kiss of peace" as practiced in the Christian liturgy was customarily exchanged for the first several centuries, not mouth to cheek, but mouth to mouth (note that men were separated from women during the liturgy) for, as the primary sources also show, this is how early Christians believed Christ and his followers exchanged their own kiss. For example, In his Paschale carmen (ca. 425-50), Latin priest-poet Sedulius condemns Judas and his betrayal of Christ with a kiss thus, "And leading that sacrilegious mob with its menacing swords and spikes, you press your mouth against his, and infuse your poison into his honey?" The kiss of peace was known in Greek from an early date as eirḗnē (εἰρήνη) ("peace", which became pax in Latin and peace in English). The source of the peace greeting is probably from the common Hebrew greeting shalom; and the greeting "Peace be with you" is similarly a translation of the Hebrew shalom aleichem. In the Gospels, both greetings were used by Jesus - e.g. Luke 24:36; John 20:21, 20:26. The Latin term translated as "sign of peace" is simply pax ("peace"), not signum pacis ("sign of peace") nor osculum pacis ("kiss of peace"). So the invitation by the deacon, or in his absence by the priest, "Let us offer each other the sign of peace", is in Latin: Offerte vobis pacem ("Offer each other peace" or "Offer each other the peace").