First we need to understand what the "Deep Magic" is/represents, before moving to the "Deeper Magic." We know from chapter 13 of The Lion, the Witch, and the Wardrobe that the Deep Magic is written in several places (on the Stone Table, on the Scepter of the Emperor-beyond-the-Sea, etc.). We also know from that chapter that it defines what Justice requires.
This closely parallels, then, the Old Testament Law, which establishes both laws and a sacrificial system for dealing with those who break the law. Colin Duriez summarizes that the Deep Magic is:
the moral order by which the world of Narnia is made and sustained. One of its inner laws is that unless the Witch is given blood for every treachery committed all Narnia will fall apart, perishing in fire and water. The Deep Magic has affinities with Old Testament law. (A Field Guide to Narnia, 181)
So, what about the "Deeper Magic"? Paul Ford likens it to "self-sacrificing love" (Pocket Companion to Narnia, 214) and points to LWW chapter 15, where Aslan says:
Though the Witch knew the Deep Magic, there is a magic deeper still which she did not know. Her knowledge goes back only to the dawn of time. But if she could have looked a little further back, into the stillness and the darkness before Time dawned, she would have read there a different incantation. She would have known that when a willing victim who had committed no treachery was killed in a traitor's stead, the Table would crack and Death itself would start working backward.
This deeper magic is thus both older and less known than the Deep Magic. This parallels in Christianity the mechanism that makes it possible for Jesus Christ's death on the cross to save (or, more specifically, atone for) law-breakers. Lewis is purposefully vague when it comes to the exact function of this mechanism (as shown elsewhere, he did not personally commit to any particular theory of atonement).
Broadly speaking, however, the deeper magic parallels God's grace in the New Covenant: a grace that was only foreshadowed in the Old Testament, a grace that fulfills the Law and its sacrificial system, a grace that was established long before the Old Testament law was formally given. Thus Duriez writes:
[The deeper magic is] a deeper principle than the natural moral order that sustains the world. This is a principle resembling the New Testament notion of grace, which fulfills and perfects the older law. (A Field Guide to Narnia, 181)