The commentaries I consulted did not have an authoritative answer nor a convincing suggestion. I suspect that what was being referred to is "scrying":
One looks intently at any of a variety of reflective surfaces such as a metal cup, a crystal ball, a chicken's liver, obsidian stone (such as arrow heads), a pond, etc. and interprets the obscure reflections therein.
Since Joseph had earned a reputation of accurately interpreting dreams he may have been esteemed as a diviner and used his reputation to lend credence to his fabricated story to his brothers:
Jamieson-Fausset-Brown Bible Commentary
5. Is not this it in which my lord drinketh—not only kept for the governor's personal use, but whereby he divines. Divination by cups,
to ascertain the course of futurity, was one of the prevalent
superstitions of ancient Egypt, as it is of Eastern countries still.
It is not likely that Joseph, a pious believer in the true God, would
have addicted himself to this superstitious practice. But he might
have availed himself of that popular notion to carry out the
successful execution of his stratagem for the last decisive trial of
Micah 3 seems to suggest that some prophets in Israel were "diviners":
NIV Micah 3: 5This is what the Lord says: “As for the prophets who
lead my people astray, they proclaim ‘peace’ if they have something to
eat, but prepare to wage war against anyone who refuses to feed them.
6Therefore night will come over you, without visions, and darkness, without divination. The sun will set for the prophets, and the day
will go dark for them.
7The seers will be ashamed and the diviners disgraced. They will all cover their faces because there is no answer from God.” 8But as
for me, I am filled with power, with the Spirit of the Lord, and with
justice and might, to declare to Jacob his transgression, to Israel
Micah does not condemn the practice explicitly but he does attribute his own success to power and virtue from God.
Please see this article in the Jewish Encyclopedia.