It's regrettable that systematic theologians used an obscure word to classify Apophatic Mysticism. Apophatic Mysticism is rooted in humility about what we can know and the realization that all of our systems are, after all, completely unable to do justice to an understanding of God. Its basic premise is that, no matter how great the human mind, it simply cannot grasp God in all His fullness, all His glory, all His might. This view is grounded in Scripture, reason, and the history of the church. Scripture's teaching that we do not yet know fully also has implications for systematic theologies.
Some Basic Writings
The foundational work of this branch of mysticism is Concerning Mystical Theology, for which the author is disputed but is known as Dionysius the Areopagite or Pseudo-Dionysius, the name or inspiration being taken from Acts 17:34. One source for this branch of mysticism is Paul's evangelistic speech in Athens in which the author's namesake Dionysius first believed (see Acts 17), where Paul notices that there is a statue dedicated "TO AN UNKNOWN GOD." The idea of "the unknown God" resonates in that there is always more to God than we have grasped; that is the starting point of this conversation. In justice I would like to point out here that Paul said to the Athenians, "What you worship as unknown, I will now proclaim to you." He then proclaimed the transcendent God who created heaven and earth, and proclaimed Christ. The "unknowing" approach, by itself, is incomplete. Still, it is easy for us to appreciate that God is beyond us. There is a part of us that ponders what is still unknown.
The most familiar modern work that treats this subject is probably The Mystical Theology of the Eastern Church, in which Vladimir Lossky proves that the Western church does not have an exclusive on dry and tedious approaches to theology. Lossky's work offers some good insights while the language is sometimes thick theologianese. For the present topic the chapter "The Divine Darkness" is worth the read.
But after all that acknowledgment of what we do and do not know, what exactly is apophatic mysticism? The apophatic approach fully respects the hiddenness of God. It reviews each image of God we try to construct in our minds and recalls: this is not God. It is a conscious acknowledgment that our image will always fall short of God and, if it becomes our focus, is then an idol which separates our worship and our thoughts from God. Some of us may think of idols as childish images of God like "the bearded old man on the throne." True enough that this risks idolatry in its way. But our systematic theologies also run the same risk of idolatry if we focus on the system, the image of God that we have built in our minds, rather than on God Himself. These systems likewise run the risk of being wrong images of God which separate us from him rather than leading us to him. Our mental images or artwork -- or systematic logical images -- can point us in the right direction if they reflect God and if we remember that they are not fully accurate pictures of God. The apophatic approach looks at each thing and consciously reminds itself, "Neither is this image fully like God." Ignorance leads to humility and to continual striving for better knowledge. Assumption of knowledge has the opposite effect.
But talking about apophatic theology like this gives the wrong idea that it is merely an intellectual honesty about the limits of what we know. While this is both necessary to spiritual health and true, it is not a complete picture. This branch of mysticism recognizes the awe and reverence for God himself that comes from a sustained focus on his status as "the unknown God" who is beyond what our minds can grasp. We look at all of our approaches to God and see how they are like God -- and set them aside as we see them fall short. After one false image after another is considered, pondered, and set aside as inadequate, there is nothing left but God Himself. As one of my favorite 19th-century mystical songs put it, "Most ancient of all mysteries, before thy throne we lie." (Federick William Faber)
Limitations and Perspective
If the "unknowing" approach becomes an exercise in denying what we know then it has gone too far. If it remains an exercise in acknowledging that "we know in part, we prophesy in part ... now we see through a glass, darkly" then it is a healthy corrective to our boasting of having a complete knowledge (or complete "systematic theology" if you'd rather). Paul says our knowledge is incomplete; it follows that a system cannot have filled in all of the blanks, organized and categorized everything we would wish to know. This is no complaint against rigorous knowledge; it is honest acknowledgment of what has been revealed and what has not; it makes careful study all the more necessary.
This type of approach, like all focuses on the hiddenness of God, must be balanced by a return to how God has revealed himself: in creation, in Scripture, and most directly in the Incarnation. The way in which we know God most fully is through Christ. The apophatic branch of mysticism, like many other parts of Christian life, cannot stand alone, but makes a valid contribution to our understanding.