"Ascension", in a simple & spiritual sense is very similar to the Eastern concept of Enlightenment found in Buddhism. To become Enlightened can simply mean to have "full comprehension of a situation". Enlightenment is a spiritual revelation or deep insight into the meaning & purpose of all things, communication with or understanding of the mind of God or Source, profound spiritual understanding or a fundamentally changed consciousness whereby everything is perceived as an interconnected unity.
Ascension in its basic spiritual or mystical sense can be thought of as the highest state of hUmanity. It is the expansion of awareness. It involves the realization of being ONE with the Creator & all of creation. An individual is 'ascending' in the sense that something of a lower vibration is ascending; it is being 'raised up' or becoming higher in vibration. This is why many Spiritual traditions aim to achieve the dissolution of the human ego (lower self) in order to realize the true nature of their being, or Higher Self.
Ascension, in this spiritual sense of the word, does not mean that one's physical body is lifted off the ground & flies through the air, ascending up into or above the clouds towards a physical place called 'heaven', although some religious adherents still literally imagine this very scenario. But for the sake of understanding, if Ascension did actually involve entering heaven without dying, then it would be more accurate to say that Ascension is about 'bringing heaven to earth' by raising the 'lower vibrational aspect of your non-physical being to a higher vibrational state of being or realization.
Ascension can also be understood in a more metaphysical way by considering the concept of 'Dimensions' or 'planes of existance, or planes of conscious'. This would mean that Ascension is about shifting from one dimension, frequency, plane of existence or consciousness (such as the 'lower' physical Earth plane) to another plane, dimension or state of consciousness that is of a 'higher frequency'.