purification: the process that restores someone to the clean standing worship requires, and the cleanness that process secures
This word belongs to the world of Israel's worship, where a person or an object was either fit to come near God or temporarily unfit. Being unfit was usually not a moral failing and almost never a matter of dirt: a skin disease, a bodily discharge, or being near someone who died all put a person out of bounds for a while. This noun names the way back in.
Nearly every use is tied to a schedule. The text speaks of "the day of his cleansing" or counts out seven days or eight, so the word covers a stretch of time and a set of required acts, not a single moment. Washing is part of it, but only part: a man with a discharge washes his clothes and bathes in fresh water and still counts seven days before he is clear (Leviticus 15:13).
A smaller group of verses turns the word outward, from a person's recovery to the standard itself. The Levites are put in charge of the cleanness of the holy things, worshippers at Hezekiah's Passover eat without meeting what the sanctuary requires, and temple staff keep watch over the cleanness of the place. Here the word is less about a process someone goes through and more about the condition holy things and holy service are held to.
The verb of this family (טָהֵר) is the act of becoming or making clean, and the adjective (טָהוֹר) describes the finished condition as a quality. This noun sits between them, naming the passage from one condition to the other and the cleanness it wins back.