What government did Jesus live?

Answer 1

In various parts of the world there were all sorts of different types of government.

The area where Jesus lived and taught was in a state of political flux in the first centuries BC and AD. By Jesus' time it was in the buffer zone between the Roman and Parthian Empires, and it varied from being a not-quite-independent "client" kingdom (under Roman sponsorship but not occupied by Roman soldiers and administrators), through being a collection only-nominally-independent "tetrarchies" sometimes under Roman occupation, to being an outright province of the Roman Empire.

According to the Gospel of Matthew Jesus was born while Herod the Great was king of Judaea. Herod was a "client king"-installed on his throne by Roman armies, but more-or-less free to run the place as he liked so long as he didn't cause trouble.

Herod died in 4 BC, and the Roman emperor Augustus (as executor of Herod's will) divided his kingdom among his three sons, who got the title "tetrarch". The area of Jerusalem was in the tetrachy of Judaea, which was ruled by Herod Archelaus until 6 AD, when the Romans annexed it as a province. But Galilee (the area where Jesus grew up and John the Baptist preached) was ruled by Herod Antipas until he was deposed by the Emperor Caligula in 39 AD. It's hard to be sure of the date of Jesus' death, but it certainly happened while Pontius Pilate was prefect of Judaea, so no later than 36 AD: Herod Antipas was still tetrarch of Galillee and Perea.

So, during Jesus' adult life, and at the time of his death, the area in which he lived and taught was divided between the Tetrarachy of Herod Antipas (a Jewish monarchy, in the northern parts, with its capital at Tiberias) and a province of the Roman Empire (in the southern parts, including and ruled from Jerusalem).

Answer 2

It depends which country or region you are referring to but most were absolute monarchies of one type or another.