The biblical scholar Matthew Henry (amongst others) notes that the verse of scripture this prayer is based on (Numbers 6:24-26) foreshadows the Trinity revealed in the New Testament. The Lord is mentioned three times, and on each occassion, with a different type of blessing. The first concerns the Lords protection, associated with God the Father. The next blessing concerns God's face (in Christ) being gracious (forgiving sin on the cross at Calvary). The final line is a peace blessing. Here this may be compared with the Holy Spirit in the New Testament, which descended on Jesus like a dove. (Matthew 3:15-17). God's peace ("Shalom") is not essentially about "quiet" or the absence of noise, but about the gift of wholeness, health and well-being.
This blessing is known in Hebrew as the "Nesiat Kapayim" and is performed in Jewish synagogues by the Kohanim. These are priests who are direct descendants of Aaron (the older brother of Moses).
The text is a quotation from Numbers Chapter 6:-
The LORD bless you
and keep you;
the LORD make his face shine upon you
and be gracious to you;
the LORD turn his face toward you
and give you peace.
(Numbers 6:24-26, NIV Bible)
In 1970, archelogists uncovered a silver amulet with this blessing etched upon it. The find is thought to date back to the seventh century B.C., making this the oldest known remanant of scripture.