What happened to the Israelites between Solomon's rule and the building of the second temple?

Briefly... the kingdom split into two "houses" or "kingdoms." The ten-tribe House of Israel, led by the two tribes of Joseph, Ephraim and Manasseh; and the House of Judah [the Jews], that was mainly composed of Judah and the tribe of Benjamin, but was populated with the priestly tribe of Levi and remnants of the others.

The Levites joined the House of Judah when Israel's king Jeroboam banished the priests from Israel, fearing that they would influence his people to return to God, Jerusalem and the kingdom of the Jews. Then, he made priests of those in his kingdom of the lowest stature, established his own religious observances and "holidays," based on those of the pagan nations around him, gave the people idols to worship and declared them to be their new gods. [see I Kings 12]

This resulted in the House of Israel angering God, losing their identity as a people, following pagan Gentile customs and traditions, and being defeated by their enemies and removed from their homes and land... never to return... lost to world history.

The House of Judah stumbled along for approximately another 130 years after Israel disappeared, before their straying from the ways of God caused them to suffer the same fate as their missing brother tribes. They suffered a series of prophetic warnings and events that brought about their captivity by the Babylonian Empire for 70 years [see Jeremiah], before they would be allowed to return to their land to rebuild the temple.

With the Jews' captivity by Babylon... the apparent throne of David disappeared from world history also [the throne that the Messiah is to inherit -- Luke 1:32]... and no Jewish king ever ruled in Jerusalem again, to this day.