No expert in Covenant Theology, I nevertheless undertake to explain it, at least my understanding of it.
This is how I would begin: imagine a person reading the Old Testament and noticing the prominence in it of the law. That same person then reads the New Testament and notices the Apostle Paul’s strong language against the law. The person reading understands the issue is salvation and so he draws the conclusion that in the Old Testament man was saved by keeping the law, and in the New Testament man is saved by grace through faith in Christ.
Theology helps us to see that this view would be disastrous to our faith. The problem with that view is that the human condition has not changed from one testament to another. We are born to the same condition that Cain and everybody born in the Old Testament was; we are born sinful. The incarnation has not changed that, nor has redemption for that matter; we are still born sinful. Both in the Old Testament after the fall and in the New Testament the problem is the same, so the solution has to be the same.
The way Covenant Theology explains how salvation in Christ is available in the Old Testament is to posit an agreement. This agreement is between God and his people, and it is called the Covenant of Grace. In the Covenant of Grace conditions are offered by God, which, if met, will provide salvation and life. The Covenant of Grace seems logically to demand a prior covenant, one that fits the circumstances of the human condition before the fall. This is called the Covenant of Works. In this covenant, God provides life to his people if they will meet his conditions. The conditions are one condition: of the tree of the knowledge of good and evil thou shalt not eat. We know the condition was not met.
I personally find the Covenant of Works problematic. I do not think it is necessary to explain the situation before the fall as a test or even as requiring an agreement. The idea of conceiving all of man’s relations to God federally does not, in my mind, necessarily always obtain. You can have seminal headship, for example. The tree in the garden does not have to be a test, it could be a reminder that man shall not live by bread (fruit) alone, but by faith in his Creator—to whom he must always relate by faith since man is finite, God infinite. I, however, know that theology does not come without its compelling logic. While remaining wary of the Covenant of Works, I tentatively affirm that in a way it makes sense. I dimly sense that the Covenant of Grace demands it.
There is another covenant, and it is the one I’m most enthusiastic about. This is the Covenant of Redemption. This covenant is an agreement between the Father and the Son. The Son agrees—in the terms of the Covenant of Redemption—to provide a representative for the Covenant of Grace. The representative for God’s people in the Covenant of Works was Adam. The representative (or federal head, hence another term for Covenant Theology is Federal Theology) in the Covenant of Grace, who meets the conditions and obtains salvation and life for God’s people, this representative is Christ. Christ is appointed to this position by the Covenant of Redemption. So Christ is the second party in both covenants, but in different capacities, as it were. In the Covenant of Redemption the Second Person of the Trinity agrees to join a human nature to the divine nature without mixture or confusion or diminution of either by means of a personal union. A descendant of Adam, Abraham and David is conceived miraculously of a virgin by the Holy Spirit, is born as a second Adam—like us in everything but sin—lives a life of obedience in a sinful world, and by his active and then passive obedience fulfills the terms of the Covenant of Grace. The Son lives and dies a human life and death, operating under the terms of the Covenant of Redemption, in order to obtain salvation for God’s people according to the terms of the Covenant of Grace.
As an aside, Limited Atonement, as you can gather, is built in.
The Covenant of Redemption, then, is between Father and Son. The Covenant of Grace, is between God and his people, with the Son being the representative of God’s people. The Covenant of Redemption is the one in which the Son is appointed and which stipulates how his conduct of that representation is carried out. The Covenant of Grace is how we are and any human ever is saved.
These theological covenants are not identical to the covenants we read about in Scripture, but are the framework behind them, as it were, their Platonic ideals (exegetes tend to be more Aristotelian and theologians more Platonic about the relation). The theological covenants are elaborated to harmonize the diverse teaching of Scripture so as to prevent us from interpreting the Bible in ways that lead to contradiction or heresy. Moreover, Covenant Theology has served as a bulwark for Protestant soteriology for centuries, providing a way of understanding difficulties and rejecting heresies in a deep and thorough and even admirable way, whatever one’s attitude to its relative merit.
That’s not that complicated, is it? At least, that’s how I understand the general outline.