to hold and exercise the authority of a chief: to govern as one placed in command over others
This verb names the activity that belongs to a chief or official. It comes from the same family as the common noun for such a man (שַׂר), the ranking figure a king sets over an army, a province, or a palace staff, and the verb describes him doing his job: holding the top place over a defined group of people and making decisions for them. The noun is everywhere in the Hebrew Bible; the verb is rare, appearing in only four verses.
Across those verses the word covers three situations that share one idea. Wisdom says that by her "princes rule" (Proverbs 8:16), and Isaiah promises a day when rulers will govern with justice (Isaiah 32:1): here the authority is legitimate and its exercise is good. In Esther the Persian decree orders that every man be master in his own house (Esther 1:22), which moves the same idea from the palace to the household. In Numbers, Dathan and Abiram fling the word at Moses as an accusation, charging him with setting himself up as their ruler (Numbers 16:13).
That last use shows what the word does and does not say by itself. It reports that someone occupies the commanding place, but it does not settle whether he has any right to be there. Hebrew has other verbs for other angles on ruling: reigning as an actual king (מָלַךְ), holding dominion or mastery over someone in general (מָשַׁל), and ruling with a firm hand over the subdued (רָדָה). This one keeps its eye on the rank of the chief and the act of filling it.