Mysticism

Mysticism involves communion with the divine, either through esoteric knowledge or directly through revelation or meditation. Mystics come in all flavors, from the deeply sincere to the outright fraudulent.

Direct communion can occur in many ways, from speaking in tongues or other forms of possession to visions and revelations. Indirect methods include tarot cards and the practices of the Kabbalah. While all mystics believe in something divine, not all believers are mystics.

Examples
Mysticism is present in countless religions across the world, with widely different doctrines and approaches. In Islam, the most prominent form of mysticism is Sufism. In Judaism, the most prominent form is the Kabbalah. Many branches of Christianity frown on this kind of direct communion, instead preferring rational argumentation and discovery of God through texts. However, Eastern Orthodox Christianity and, to a lesser degree, Roman Catholicism employ mysticism in theological discourse. Hinduism, Buddhism, Sikhism, and Taoism, just to name a few, are some non-Abrahamic religions that have mystical practices. Tantra is arguably mysticism, as well. Shamanism in many folk religions around the world could also often be described as mystical in nature.

Scientific explanation for ecstatic experiences
Religious ecstasy is an altered state of consciousness that is often experienced by people practicing mysticism. One frequently finds that mystical experiences get induced by stressing the body (through hunger, thirst, sleep deprivation...), and/or by stressing the mind through isolation. All this can cause hallucinations, as can drugs that are used to induce mystical experience. There is every reason to suspect that mysticism - occurring across multiple cultures and across multiple religious traditions - is due to natural reactions in the brain.

In D&D
Mystics are also a character class in D&D, at least in good ol' Second Edition, and are there a type of cleric who forgoes the use of magic and most types of weapons in exchange for witchcraft-like abilities as brewing potions or magical candles to get a trip and fly higher than a kite, er, dragon.