Hellenization is the interaction between Greek culture and native cultures. By the time of Jesus, all forms of Judaism (and there were many different forms) could be classed as "Hellenistic Judaism," because, whether through active assimilation, passive assimilation, or resistance, they were all changed by the encounter with Hellenistic culture.
Some effects of Hellenization were:
1) Widespread use of the Greek language
2) Spread of Greek religion (various cults - Apollo, Athena, Dionysus, mystery cults, etc.)
3) Spread of Greek philosophical thought (i.e., the philosophy of Plato, Aristotle, etc.)
4) Greek views of the body. The male body was the perfect object; circumcision detracted from the body's perfection.
5) Compared to many Jews, there was a laxity in food laws.
6) Gymnasiums - schools in which boys were formed in the culture of the Greek world.
7) Athletic events - athletes competed naked.
Why was this seen as bad? Many aspects of Greek culture conflicted more or less directly with various aspects of the Jewish law. For some Jews, the acceptance of Greek culture was seen as impious and unrighteous.
Hellenism