These guys lead the first group of Jewish exiles back to Judah, lay the foundations for the second temple, and build the altar. However, their non-Israelite enemies temporarily disrupt the rebuilding, since Zerubbabel and Jeshua refuse to let them participate in the building. The enemies accuse the Jews of being a historically rebellious people, telling the Emperor Artaxerxes that they shouldn't be allowed to rebuild.
But stoked by Haggai and Zechariah's prophecies, Jeshua and Zerubbabel rebuild anyway during Darius's rule. They tell the governor, Tattenai, that Cyrus gave them permission to build; they can look it up. Darius discovers that this is true and lets them continue, providing them with the resources they need.
The name Zerubbabel means "Sown (or born) in Babylon." Jeshua or Yeshua in Hebrew is the same name as Jesus and basically the same name as Joshua. It means, "God is salvation."
Judea was a very small, relatively insignificant part of the Persian empire after its conquest of Babylon. It was impoverished, depopulated, and probably unstable, being surrounded by hostile tribes. The Persian kings are portrayed as pretty magnanimous in their treatment of Judea, but it was smart imperial policy. Happy provinces are loyal provinces, and the kings didn't want to have any local skirmishes in the boondocks. They had more powerful countries like Egypt and Macedonia to worry about.