priesthood: the office, ministry, and standing of a priest before God
This word is the abstract noun for the priest's office. Where כֹּהֵן names the man who serves, this word names the role he holds: the sacred service he performs, the rank he belongs to, and the God-given right to draw near to the altar. It appears most often in the laws of Exodus and Numbers, where the office is granted to Aaron and his sons "by a perpetual statute," and it turns up again in Ezra and Nehemiah, where belonging to the priesthood had to be proved from the family records.
Across its uses the word bends in three directions that all share one core. Usually it points to the office and its ministry itself, the service and the lasting covenant grant of that service to one family (as with the covenant of a "perpetual priesthood" given to Phinehas for his zeal). Sometimes it points to the priesthood as a hereditary body a person is counted in or shut out of, as when returning exiles who could not document their descent were barred from it. Once, in Hannah's story, it narrows to a priestly post sought simply as a way to earn bread.
The Greek of the New Testament carries the same idea with ἱερωσύνη, the word Hebrews uses when it argues that Christ holds a priesthood that does not pass away.