What was the conflict between the Hellenists and the Hebrews about? How was it resolved?

Answer 1

I assumed you're talking about the Chanukah story? It was a matter of philosophy and religion.
The Jews were more spiritually and mystically oriented. They believed that there is a world beyond our five senses, that there is one benevolent and involved God, that not everything is or chould be understood by human beings, and that beauty is a good thing, but only if you use it to further spirituality.
The Hellenists (Greeks) were very materialistic. They believed in physical beauty, human logic, and what could be directly percieved by the senses as the be-all and end-all of life. They also had a firm conviction that anyone who disagreed with their culture must be a total moron.
Adding insult to injury, the religion that they had was polytheistic.

So the Greeks tried to make the Jews Greek (and succeeded with a large portion of the population). The more religious Jews stubbornly stayed Jewish. The Greeks started passing laws outlawing Torah study and Jewish rituals, and defiled the Temple in Jerusalem.
If you go on Aish.com, under Chanukah, they have a bunch of good articles about the whole philosophical conflict. Now on to the resolution:
A family of priests called the Hasmonians/Macabees started a revolt against the Hellenists. Led by Yehuda Macabbee, they beat back the Greek army and captured Jerusalem. Immediately, they went to work cleaning up the Temple.
That's where the Chanukah menorah comes in. One important part of the daily service was that the huge candelabra called the menorah had to be lit every day. There was only enough oil to last for one day. They lit it anyway, and it stayed burning for eight days, which was how long it took to prepare new oil.
Some time later, the Greeks retook Jerusalem and killed four out of the five Maccabee brothers. I'm not exactly sure what happened after that, except that we Jews are still here, and the ancient Greeks aren't.

Answer 2

The Hellenists believed the Hebrews to me annoying the Gods and bringing their wrath down. It was only resolved when the Hellenists ran the Hebrews as far away as possible.