A covenant ("Brit" in Hebrew)is a promise or oath. The word "covenant" is used hundreds of times in some English translations of the Bible to describe God's promises to his people.
In Hebrew, the making of a Covenant, is called the "cutting" of a Covenant - "Llikhrot brit" while the fullfilling of a Covenant is "lehaqim Brit". Cassutto points out that this terminology is misunderstood by some higher critics of the Bible who believe that the difference in the Hebrew expression implies different authors coming from different times, and which support the documentary theory of biblical origins.
"Signs of the covenant" are given properly for the eyes of God and not given for pedagogic purpose for the sake of the people. The people of the Lord lift up or fulfill the signs of the Covenant to heaven itself. Heaven (God) looks down and sees the signs, sees the faith of the people that are under the sign, sees them taking refuge in what the signs represent, so that the people are found safe under the wings of God's grace and mercy. The signs of the New Covenant are thus analogous to the Old Covenant signs that God looked upon - the blood of the lamb at the first Passover which was placed by the people who believed, which was seen by the angel, and so the people were spared; the circumcision of the Israelites at eight days of age, which Moses neglected to do for his own son, and was spared only because of the faithful love and determination of his wife who did the job for him; and the rainbow in the skies, which, when God would look at, He would remember His promise to spare the earth from more destruction by water as in the days of Noah. These are all echoes of the Jewish "Remembrance" prayers from the time of Jesus and thus provide the setting for the celebration of the Eucharist.
The term is also used in other contexts, such as covenant marriage (where by law a married couple agrees to counseling before seeking divorce) or "covenant membership" (where a church member must seek permission of the elder body before resigning his/her church membership).