Short answers, pulled from the story.
Christian mysticism is defined by scholar Bernard McGinn as that part of Christian belief and practice that concerns the preparation for, the consciousness of, and the effect of a direct and transformative presence of God. McGinn prefers the word "presence" over "union" because not all mystics spoke of union with God, and because the tradition emphasizes transformation in both the mystic and those the mystic has affected.
The three stages formalized by Evagrius Ponticus (345-399 AD) and Pseudo-Dionysius are catharsis or purification, theoria or illumination, and theosis or union with God. In Western churches these became the purgative, illuminative, and unitive ways; in Eastern churches they became prayer of the lips, prayer of the mind, and prayer of the heart.
The Jesus Prayer is a short formulaic prayer: "Lord Jesus Christ, Son of God, have mercy on me, a sinner." It began in the early church as a training exercise for spiritual repose and was later developed by Byzantine hesychasts into a full practice with technical requirements. It is described as the first stage of noetic prayer in the Eastern Church, leading toward theoria, the vision of God.
The three most prominent Spanish Christian mystics were Ignatius of Loyola, Teresa of Avila, and John of the Cross. Teresa organized mystical prayer into nine grades and described the Prayer of Quiet using the image of four ways of watering a garden. John of the Cross developed the concept of the two dark nights - the dark night of the senses and the dark night of the soul - as stages of renunciation on the way to union with God.
Theosis, also called deification or divinization, is the ultimate goal of the Christian life in Eastern Orthodox teaching. According to John Romanides, it means attaining likeness to or union with God. A phrase attributed to Athanasius of Alexandria captures the idea: "God became human so that man might become god." It is achieved through contemplative prayer and the cultivation of watchfulness, or nepsis.
Cataphatic theology approaches God by affirming what God is, using images, words, and imagination; its practitioners include Ignatius of Loyola, Julian of Norwich, and Francis of Assisi. Apophatic or negative theology approaches God by negating all concepts, resting in imageless stillness; it was inspired by the writings of Pseudo-Dionysius the Areopagite (late 5th to early 6th century) and forms the basis of Eastern Orthodox hesychasm, while also influencing Western Catholic mysticism from the 12th century onward through works like The Cloud of Unknowing.