When Jesus was born that part of the world was under the Roman Empire, locally it was called Judea and it was part of the larger province of Siria

As such it would have a governor appointed by the Roman Senate, but it's also worth noting that Romans had the habit of incorporating local government structures into their own government. If a tribe had a king for example the governor of the province would work alongside that king and coordinate to collect taxes, build infrastructure, and so on (reminiscent to the way many modern african nations incorporate tribal governments come to think of it). Of course sometimes local rulers would not cooperate but that's besides the point.

In the case of Judea there was Herodes, who was officially the king of the jews, although he was not actually a jew and his claim to the throne was contested to say the least. It would be easy to get lost in the weeds here trying to explain who he was, so I will avoid it, but if you are interested you should read about his life

Along with the King of Jews there was also a sort of unofficial council made by jewish religious leaders, the Pharisees. They didn't have political power exactly, but they had religious power, and people would often listen to them, so I think they were part of the government in practice even if not officially.

The Pharisees had a love hate relationship with Herodes. At times they seemed to hate him but at other times they would cry in his parties telling him that he was indeed a jew... it's complicated

Anyway, along with the Roman governor, the Jewish King, and the Pharisees there would be many other roman government officials in charge of smaller territories, like Pontius Pilate, who was the governor of Judea when Jesus was crucified

There were also many tax collectors, generals, judges and many many different kinds of public servants. Most were part of the roman bureaucracy, but some were part of the institutions the romans chose to preserve, in this case mainly religious leaders

Although I should mention that all of these non roman institutions came crashing down shortly after Jesus's death when the jews revolted against the empire, which culminated with their expulsion from Judea, the destruction of the temple, and the destruction all their government structures like their King or the Pharisees... although the Pharisees survived in a way and created modern judaism