By around 320 BC, Alexander the Great had conquered the whole Mediterranean world as well as most of the Middle East. This brought Greek language and culture to Israel and Palestine. When Alexander died, most of the Middle East was divided up between his generals Seleuca and Ptolemy, whose descendants became known as the Seleucids and Ptolemies. One of the Seleucid rulers, Antiochus Epiphanes, wanted everyone in his kingdom to share the same culture and gods. So he began to squash Judaism around 170 BC. Any Jew who would not worship Greek gods was put to death. Their scrolls were burned, their sabbath day became illegal, their temple in Jerusalem was desecrated, and pigs were even sacrificed on their altar.
This really ticked off the Jews at that time, and it began a resistance movement known as the Maccabean Revolt. Led by Judas Maccabeus (the Hammer), the Maccabean Jews took on the Romans equipped only with spears, bows, arrows, and rocks. Although greatly outnumbered, they showed great determination and grit, and somehow were able to take back the temple in Jerusalem in 164 BC. The first thing they did when they entered the temple was to light a makeshift menorah, but they could only find one vial of the pure oil to be used. They used the small amount of oil to light the menorah, and miraculously it stayed lit for 8 days. The retaking of the temple and the miracle of the menorah is what is celebrated on the Jewish holiday of Hanukkah.
The brother of Judas Maccabeus, Simon, later became high priest and ruler of Israel. He was a type of king, but he couldn't really be the king because only someone from the line of David could be king. Simon was from Aaron's line, which was the priestly line. Simon and his descendants became what is known as the Hasmonean Dynasty, which lasted for around one hundred years. King Herod would come later at the end of this dynasty, a ruler of Israel, but not a true king.
The Greeks would give way to the Romans in 63 BC, when the Roman General Pompey took over the city of Jerusalem. This brought Roman administration, Roman law, Roman roads, and Roman peace, the "Pax Romana". A few years later, a young man named Octavion would come to power in Rome, who would later be known as Caesar Augustus.
And with that, the stage was set for the coming of the Messiah, the true King of Israel.
"When the fullness of time had come, God sent forth His Son, born of woman, born under the law" - Galatians 4
"And it came to pass in those days, that there went out a decree from Caesar Augustus that all the world should be taxed." - Luke 2