A sort of peak experience during which one sees God, intuits the oneness of the universe, connects with tradition or history, and the like. These can be mystical, and are generally assumed to be such. Usually ecstatic. If you're born again into Protestant Christianity, that's a sort of religious awkening that dates back to the great awakening of the early-middle eighteenth century, led by such ministers as Jonathan Edwards. You can have other sorts as well, though, including Buddhist, Taoist, Hindu, Jewish, and Pagan religious awakenings.
These are generally, and should always be life-altering experiences. Some claim that there is a chemical basis for them, but you never know.
(followup) It's clear to me that these experiences are always understood within the existing psychological system possessed by the person having them. That is, the individual interprets the experience in some way as to try and make sense out of it. You can't do that in a vacuum.