Mysticism and "Awakening"
The term 'mystic' comes from the Ancient Greek mystikós, from mýstēs1, originally referring to an intiate in ancient "mystery religion," which were secret (at least in the sense that their practices were secret) cults in Ancient Greek and Rome. Mýstēs itself is from a word meaning "shut," and further both "hidden" and "secret." The term was used by Pseudo-Dionysius, an early church father writing in the 6th century, in describing a theology and practice of union with God, drawing on an earlier Greek pagan mystical tradition.2
In practice, both in academic work and in modern mystical writing, 'mystical' is used in varied enough ways that it doesn't communicate a lot narrowly, out of context. This is obviously letting alone the colloquial usage which doesn't mean a lot more than "magical" and "spooky." In general it tends to mean something like "union with God," or "gnosis of ultimate reality beyond ordinary apprehension", but this bleeds into what's more often known as "occultism" which tends to have more to do with ritual magic and interpretation of symbols. "Mysticism" is also often used to refer generically to the practices and experiences of altered states or even just unusual perception, achieved by religious or spiritual practice or psychedelics.
"Mysticism" is more meaningful in reference to any specific tradition, in which case we can talk about specific claims, practices, and reported experiences:
Insight
Inquiry
paths, and meditation progress
gradualism and suddenism
Footnotes
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It's maybe contentious whether Pseudo-Dionysius was drawing his whole theology from Neo-Platonism or if it was an innovation within Christianity, and maybe even whether 'mystical' came directly from either the mysteries or Neo-Platonism at all. See Louis Bouyer 1963, The Spirituality of the New Testament and the Fathers, pp. 406-407 ↩