die Herrschaft
rule
The exercise of power and control over a country, a people, or a territory.
Die Herrschaft des Kaisers dauerte über vierzig Jahre.
The emperor's rule lasted more than forty years.

rule / control
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rule
The exercise of power and control over a country, a people, or a territory.
Die Herrschaft des Kaisers dauerte über vierzig Jahre.
The emperor's rule lasted more than forty years.

'Die Herrschaft' (rule) is derived from 'der Herr' (master, lord) and is closely related to the verb 'herrschen' (to rule). All words in this family have to do with power, control, and authority.
Der König herrschte mit großer Strenge über sein Volk.
The king ruled over his people with great severity.
In historical or political contexts, 'Herrschaft' (rule) is often combined with verbs such as 'übernehmen' (to take over), 'antreten' (to assume), or 'sichern' (to secure). It describes the beginning or consolidation of power.
Nach dem Tod des Diktators endete seine grausame Herrschaft.
After the dictator's death, his cruel rule ended.
In the meaning 'Kontrolle' (control), 'Herrschaft' is almost always used with the preposition 'über' (over) and the accusative. Typical verbs here are 'verlieren' (to lose), 'gewinnen' (to gain), or 'behalten' (to keep).
Er kämpfte darum, die Herrschaft über seine Gefühle zurückzugewinnen.
He struggled to regain control over his feelings.
The form of address 'meine Herrschaften' (ladies and gentlemen) is now used only rarely in extremely formal or ironic contexts and usually sounds outdated. It is used almost exclusively in the plural.
„Meine Herrschaften, darf ich um Ihre Aufmerksamkeit bitten?“, fragte der Butler.
'Ladies and gentlemen, may I have your attention?' asked the butler.
government
'die Regierung' and 'die Herrschaft' can both have to do with political power over a country. 'Die Regierung' usually means the specific political leadership or the act of governing in a state. 'Die Herrschaft' puts stronger emphasis on power and control over a country, a people, or a territory. The word often sounds more general or historical, for example with kings, dictatorships, or long periods in power. The words can be confused when talking about political power. For the current political leadership, 'Regierung' usually sounds more natural.

Unter der Herrschaft des Königs hatten viele Menschen kaum Rechte.
Under the king's rule, many people had hardly any rights.
Die Regierung muss auf die Fragen der Bürger antworten.
The government must answer the citizens' questions.