What understanding of ethical behavior did greek philosophers embrace that still communicate develop a moral compass of what is right or wrong

That's a hard question to answer, because ancient Greek philosophers had differing perspectives on ethics, and elements of their various approaches all still are around today.

Plato's famous work, The Republic, portrays the right sort of state as a magnification of the right sort of person.  Socrates served as Plato's mentor and is featured heavily in dialogues Plato wrote.  Their view was that an ethical person is guided by pursuit of higher truths beyond the visible, material plane.  Socrates' view that an ethical man will always do what is best and honorable was in contrast to other philosophers of his day, such as Thrasymachus, who contended that "justice" or "ethics" was whatever the strongest persons in society said it was.  Thrasymachus and others of the group we call the Sophists were relativists.

Aristotle believed the ethical life was pursuing what made a human life the most flourishing it could be, and that the right (ethical) path is a mean between extremes.  So, for instance, courage is the ethical virtue that occupies the middle position between cowardice (fleeing from a conflict) and foolhardiness (rushing headlong into conflict without thinking).

Epicurus put forth the ethical theory that the good life (the ethical life) is found by doing that which results in the least pain, thus producing the most pleasurable existence.  Epicurus' ideas formed the basis of later utilitarian theory that the right path is that which produces the greatest well-being for the greatest number of people.

That's just a sample of ethical ideas that were discussed and debated in Greek philosophical thought.