In Orthodoxy and Eastern Catholicism, mysticism can be broadly understood as all the practices oriented towards contemplation of and union with the Trinity.

A few helpful terms here include theosis, apophasis, and theoria.

The first thing to understand is that Eastern Christianity understands salvation in terms of "deification," or theosis. To put it in a common formulation, deification means that human beings will become by grace everything that God is by nature. The goal of Eastern Christian practice is to be drawn up into the eternal life of the Trinity in order to participate by grace in God's own divinity.

Another common feature of Eastern Christianity is the idea of divine apophasis, or unknowability. Because conceptual thinking originates in our experience of finite realities, the infinite God always exceeds our conceptual knowledge. Therefore, the process of coming to know God proceeds by the path of unknowing, or negating our concepts of God by forming antinomies. Thus, while the Christian tradition tells us that God is good, for example, we know that there's an infinite gulf between what God is like and what we can conceive as "goodness," and therefore, the Christian mystic must simultaneously hold the propositions "God is good" and "God is not good." They would basically do the same about every positive statement about God: hold it together with its negation, always seeking higher, fuller descriptions of God until conceptual thought breaks down.

The goal of apophatic practice is theoria, or contemplation, which can be describes as a direct "vision" of God. Theoria is attained alongside theosis. In other words, we finally come to know God, non-conceptually, to the extent that we "become God" by participation in God's own life.

With these basic principles in mind, Eastern Christianity sees theology as inextricably tied to spiritual practices aimed at growth towards deification. In fact, "theologian" is an honorary title to refer to those whose writings bear witness to a direct experience of the Trinity, and it's generally reserved for three figures: St. John the Apostle, St. Gregory of Nazianzus, and St. Symeon the New Theologian.

Basically, then, when we talk about mysticism with respect to the Eastern churches, we're talking about an understanding of Christianity aimed at union and contemplation. And really, the whole of Christian practice in these churches, when they actually live up to their own ideals, are directed towards those goals. The liturgy, the sacraments, ascetic practices (e.g., following the fasting schedule), prayer, and so on, are all understood to be ways of growing towards union with and contemplation of the unknowable God who has become known to the world through the apophatic body of Christ.

This is, admittedly, an extremely cursory overview, since I'm in a rush, but I'd be happy to answer any questions you have about it.

Some rather accessible and very significant primary works on Eastern mysticism include Pseudo-Dionysius's The Divine Names, Gregory Palamas' The Triads, and Gregory of Nyssa's The Life of Moses.

Christian mysticism is simply a broad categorization of various Christian traditions that focus on encountering or experiencing the presence of God or spiritual union with Him. There are mystic traditions in Western Christianity as well, such as those in 13th century Italy (Catherine of Siena, St. Francis) and 16th century Spain (San Juan de la Cruz, Santa Teresa de Avila), though there are countless examples from all across Europe and the Near East over the course of Christian history.

In mystic literature, there is a general emphasis on contemplation, the transcendental experience, and the ineffable divine, but considering the breadth of the category, it's application and expression was and is incredibly diverse.