The Hellenistic Period, or the Hellenistic Civilization, refers to the growth and expansion of Greece’s cultural influence in the world.
It started at the death of Alexander the Great, and went until the emergence of Rome as a world superpower. Basically, what would happen is Alexander the Great would conquer an area of the Persian Empire, generally in Asia or Africa.
The transcript is for your convenience
And then, he would set up a city there. Both veterans of Alexander the Great’s army would settle on that area, and people from Greece would move to that area in the Persian empire, and live in that city as well, bringing with them Greece’s culture. Everything from arts and theater, to mathematics and science.
So, basically, they were imposing the Greek culture on the people living in that city, the Persian residence. It’s interesting to note that we don’t know what Alexander the Great’s motivation was in setting up those cities. It could’ve been to impose control in that newly conquered region of the Persian Empire, or it could’ve been to impose the culture of Greece on the people living in the Persian Empire at that time.