The government of the kings of Rome had some connection to that of the kings of the heroic times of the Greeks. It disappeared like the others because of its general failing, although in itself and in its particular nature it was very good.
To give a notion of this government, I shall distinguish those of the first five kings, that of Servius Tullius, and that of Tarquin.
The crown was elective, and under the first five kings the senate had the largest role in the election.
After the king’s death, the senate examined whether to keep the form of government that was in place. If it judged apposite to keep it, it named a magistrate  chosen from its own number who elected a king ; the senate had to approve the election, the people confirm it, and the auspices endorse it. If one of these three conditions was not met, another election was required.
The constitution was monarchical, aristocratic, and popular ; and such was the harmony of power that neither dispute nor jealousy was seen in the early reigns. The king commanded the armies, and had the intendancy of the sacrifices ; he had judicial authority in civil  and criminal  matters ; he convoked the senate, assembled the people, took certain matters before them, and decided the others with the senate. 
The senate had broad authority. The kings often called on senators to judge with them ; they did not take matters to the people unless they had first been discussed  in the senate.
The people had the right to elect the magistrates, to approve new laws, and, when the king so allowed, to declare war and make peace. They did not have judicial authority. When Tullus Hostilius returned the judgment of Horace to the people, he had his own reasons, which we find in Dionysius of Halicarnassus. 
The constitution changed under Servius Tullius.  The senate played no part in his election : he had himself proclaimed by the people. He divested himself of civil judgments,  and reserved to himself only the criminal ones ; he took all business directly to the people ; he relieved them of taxes, and put the whole burden on the patricians. Thus, as he was weakening royal authority and the authority of the senate, he was increasing the power of the people. 
Tarquin did not have himself elected either by the senate or by the people ; he considered Servius Tullius a usurper, and took the crown as an hereditary right. He exterminated most of the senators ; he no longer consulted those who remained, and did not even call them to his judgments. His authority grew ; but what was repugnant in that authority became even more so : he usurped the power of the people, made laws without them, and even made some against them.  He would combined the three powers in his person, but the people remembered for a moment that it was the legislator : and Tarquin was no more.