kingship: the royal office and the authority to reign as king
This word names kingship in the abstract. It does not point to the land a king rules or the people under him, but to the royal office itself and the power that comes with sitting on the throne. It is the thing a man wins, loses, inherits, or hands on when he becomes or ceases to be king.
Most of its appearances sit inside Israel's fiercest struggles over the throne: Saul against David, Adonijah against Solomon, the tearing of the kingdom in two after Solomon. In those stories the word behaves almost like a prize passed from hand to hand. It then rises to its highest use when a psalm and a prophet declare that the kingship belongs to the LORD, that final royal authority is his and not any earthly king's.
Hebrew also uses the word attributively, to mean simply "royal." Attached to something that belongs to the crown, it gives us the royal city, a royal turban, and above all the "royal seed," a person of the reigning line.
It is a narrower word than מַמְלָכָה, which points to the kingdom as a realm, and than מַלְכוּת, which more broadly covers a reign, its span, and its dominion.