What was the political system of the Israelite Kingdoms?

Answer 1

Hebrew society was based upon the covenant with God, as set forth in the Torah. The Torah establishes a separation of the various forms of power, so that all of them are not concentrated in one individual.
In the earliest era, the Hebrews consisted of family-groups (clans; tribes) led by patriarchs. Later, during the three centuries of the Judges, the Israelite nation had sages (led by Sanhedrin-courts) who were consulted in questions of Jewish law and conduct; while individual Judges occasionally unified the people in war against some common enemy.

Link: Jewish history timeline

During the four centuries of the monarchy, the Israelites were led by three distinct people or groups: the king, the Sanhedrin and the Kohen Gadol. (See also Deuteronomy ch.17-18.)

• The king conducted the nation and made decisions in most national matters but was not necessarily one of the leading sages. It is noteworthy that although the prophets had no temporal power, they were sent by God not only to exhort the people in general, but also to rebuke the king when needed.
  

Link: The role of the prophets

• The Sanhedrin (court of Sages) was the final authority on Torah-matters. One of its functions was to ensure that the Torah-traditions were handed down intact from generation to generation.
  

• The Kohen Gadol conducted the observances in the Holy Temple, together with the assistance of the rest of the Kohanim and Levites (Leviticus ch.21, Numbers ch.8 and 18). Though the Kohen Gadol had no formal power outside the Temple, he did have great influence due to the prestige of his position. In times of need, he occasionally dealt directly with foreign monarchs (see Talmud, Yoma 47a and 69b).
  

When the monarchy ceased at the time of the First Destruction, the prophets pointed out that the reason was that there had been some individuals, men with positions of power, who misused that power. As was the case concerning the avoidance of idolatry, not everyone heeded the prophets.
Specifically, the monarchy and its hangers-on, and the public functionaries, included a minority that took advantage of those who had no such power.

See also the other related Links.

Link: Reasons for the Destruction

Link: Social classes in ancient Israel

Link: Life in Israel

Answer 2

Hebrew society was based upon the covenant with God, as set forth in the Torah.
In the earliest era, the Hebrews consisted of family-groups (clans; tribes) led by patriarchs. Later, during the three centuries of the Judges, individual Judges occasionally unified the people in war against some common enemy.
During the four centuries of the monarchy, the Israelites were led by three distinct people or groups (See also Deuteronomy ch.17-18.)
The king conducted the nation and made decisions in most national matters but was not necessarily one of the leading sages.
The Sanhedrin (court of Sages) was the final authority on Torah-matters.
The Kohen Gadol conducted the observances in the Holy Temple, together with the assistance of the rest of the Kohanim and Levites (Leviticus ch.21, Numbers ch.8 and 18).

Answer 3

They the Jews had judges, and no king, but the Jews asked Samuel to find a king for them like the other tribes had.