What does the Roman Catholic Church teach about covenant theology?

Outline from some RCIA material on the Six Covenants

This outline is backed up by various Cathechism articles and the Catholic Encyclopedia. I have included some of the scriptural references that I have in the notes.

The Catholic Church teaches that there are six covenants. But first ...

What is a Covenant?

God’s Covenants are ways that he reveals Himself, and his Gifts, to His people.

CCC 204 God revealed himself progressively and under different names to his people, but the revelation that proved to be the fundamental one for both the Old and the New Covenants was the revelation of the divine name to Moses in the theophany of the burning bush, on the threshold of the Exodus and of the covenant on Sinai.

It took a bit of explaining from our deacon to get my head around these points. You see the Old and New Covenants referred to often, to include here in the Catechism:

The celebrants of the heavenly liturgy
1138 "Recapitulated in Christ," these are the ones who take part in the service of the praise of God and the fulfillment of his plan: the heavenly powers, all creation (the four living beings), the servants of the Old and New Covenants (the twenty-four elders), the new People of God (the one hundred and forty-four thousand), especially the martyrs "slain for the word of God," and the all-holy Mother of God (the Woman), the Bride of the Lamb, and finally "a great multitude which no one could number, from every nation, from all tribes, and peoples and tongues."

1. A Covenant (used in this older sense) can be both a Gift and a commitment
2. A key difference between a covenant and a simple contract is that a covenant in this older sense can be a one-way transaction; it does not hinge upon "offer and acceptance" in the way that a contract does, but for practical purposes the recipient will at least acknowledge it.
3. The terms Covenant and Testament in English are linked in meaning to the original Hebrew and Greek terms

(Strong's 1242 diathḗkē (from 1223 /diá, "thoroughly," intensifying 5087 /títhēmi, "place, set") – properly, a set-agreement having complete terms determined by the initiating party, which also are fully affirmed by the one entering the agreement).

It's meaning takes a little bit of digging to get a handle on.

However, at an early date, the Alexandrian translators of the Scripture, known as the Septuagint, employed the word as the equivalent of the Hebrew berith, which means a pact, an alliance, more especially the alliance of Yahweh with Israel. In St. Paul (1 Corinthians 11:25) Jesus Christ uses the words "new testament" as meaning the alliance established by Himself between God and the world, and this is called "new" as opposed to that of which Moses was the > mediator. (Catholic Encyclopedia, Testament)

There is the added nuance that ancient Mid-Eastern covenants could be of either a promissory or obligatory character.

The Five OT Covenants

One of the themes we had to emphasize as Catechists was that the History of The Church is The Record of the Covenants between God and Mankind.

1. Adam and Eve
   God revealed the beginning of humanity and gave Man and Woman to each other (Gifts from God, they were to each other, in the beginning) (Genesis, numerous passages in books 1, 2, 3)

2. Noah / The Flood;
   The covenant/gift/Promise was to never again to wipe clean the world with a Flood (Genesis 6-9) (CCC 56-58)

3. Abraham
   God revealed that all families on Earth would be blessed through Abraham him & his seed.
   “I will multiply your descendants as the stars in the heavens.” (Exodus 32:13)

4. Moses
   God freed His people from slavery (Exodus 5: 1). The great gifts of the Law (Ten Commandments) and The Promised Land were granted.

5. A Kingdom with an everlasting throne. This covenant is associated with King David. (Isaiah 9:6-7)

In each of the above cases, the people eventually fell into sin, which is by Catholic definition "a turning away" from God. The call would come through the prophets to repent, which means to "turn toward God" in that sense, although in Noah's case he heard a call to simply listen to and obey God, and in Abraham's his heart's call was to completely trust God.

The Prophecy that foretold the new Testament, the final Covenant, was that “The Kingdom will be re-established by The Messiah” (The anointed one). That's linked to the fifth OT covenant.

The New Testament (also The Final Covenant)

God’s 6th (final) covenant is that God reveals Himself in “The Word, Incarnate” and in so doing gives the Gift of his only begotten Son. (Matthew 16:15-17) This promise, this gift, is meant for all of mankind. (See Peter Turner's answer on the "scaling up" as the covenants progress). The fruits of that gift are eternal life. It is worth noting that Jesus identifies "the new covenant" in Luke 22:19-20 during the Last Supper.

19 Then he took the bread, said the blessing, broke it, and gave it to them, saying, “This is my body, which will be given for you; do this in memory of me.” 20 And likewise the cup after they had eaten, saying, “This cup is the new covenant in my blood, which will be shed for you."

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I had pages and pages of material on this, but that's a summary of what we taught.