Left: Magian with barsom (sacred twigs) from the Oxus treasury (British Museum)

1 And it came to pass when Jerusalem was taken, in the ninth year of Zedekiah king of Judah, in the tenth month, came Nebuchadrezzar king of Babylon and all his army against Jerusalem, and they besieged it.
2 In the eleventh year of Zedekiah, in the fourth month, on the ninth of the month, the city was broken into;
3 and all the princes of the king of Babylon came in, and sat in the middle gate: Nergal-sharezer, Samgar-nebu, Sarsechim, chief chamberlain, Nergal-sharezer, chief magian, and all the rest of the princes of the king of Babylon.
-- Jeremiah 39:1-3

Who the magi were -- Non-Biblical evidence

We may form a conjecture by non-Biblical evidence of a probable meaning to the word magoi. Herodotus is our authority for supposing that the Magi were the sacred caste of the Medes. They provided priests for Persia, and, regardless of dynastic vicissitudes, ever kept up their dominating religious influence. To the head of this caste, Nergal Sharezar, Jeremias gives the title Rab-Mag, "Chief Magus" (Jeremiah 39:3, 39:13, in Hebrew original — Septuagint and Vulgate translations are erroneous here).

The Conspiracy of the Magi
The Magi seize the throne of the Persian empire at Susa by passing off one of their number as Smerdis, the brother of Cambyses. The real Smerdis has been executed on Cambyses’ orders (61). At Ecbatana in Syria, Cambyses hears a proclamation in the name of King Smerdis. He questions his henchman Prexaspes, who assures him that he, Prexaspes, buried Smerdis with his own hands (62). They interrogate the herald and discover the true identity of the pretenders (63). Cambyses tries to ride for Susa, but accidentally jumps on his sword and gravely wounds himself. Thus a dream and a prophecy both come true for Cambyses: the dream that Smerdis would be King (which is what persuaded Cambyses to kill his brother in the first place) and the prophecy that he himself would die at Ecbatana (64). With his dying breath, Cambyses appeals to the noble Persians not to let the Persian empire fall into the hands of the Medes (65). But the nobles do not believe that the Smerdis who is now King is not the same man as the brother of Cambyses (66). So Smerdis the Magian rules for seven months and wins support by granting tax breaks and freedom from military service to the subject nations (67). Finally Otanes, a noble Persian, begins to suspect Smerdis. Otanes’ daughter is one of Smerdis’ wives; Otanes uses her access to Smerdis in bed to determine that Smerdis has no ears, which proves he is Smerdis the Magian and not Smerdis the son of Cyrus (68-69). A junta of seven nobles plans to overthrow the Magi (70). Darius and Otanes debate whether to strike at once, which Darius favors, or to wait, which seems better to Otanes. Darius carries the day. (71-73) Brief digression: the last days of Prexaspes. The Magi hire him to address the populace from the palace walls, and to reassure them that Smerdis the son of Cyrus is indeed King. Prexaspes agrees, but when he gets up on the walls, instead of propagandizing for the Magi he tells the people the truth. Then he commits suicide by toppling headlong from the walls (74-75). The junta approaches the palace and is encouraged by a bird sign (76). They walk right past the guards; when the eunuchs try to stop them, they kill them (77). They proceed into the inner chamber and kill both of the Magi. They then run back into the streets, killing every Magian they meet and announcing the news; soon all the Persians are busily killing Magi, and the holiday called the "Slaughter of the Magi" commemorates the event (78-79).
-- Herodotus, The Histories, Book Three (61-97)

against the wanderers of the night: the Magians, the Bacchantes, the Maenads and initiates. Heraclitus threatens them with tortures after death, he threatens them with fire, for what they believe to be initiations in the mysteries are in fact impious rites.
-- Clement of Alexandria, Protrepticus 12

It was at this time that the Persian priests, the Magians, were first established as an order, and always at break of day Cyrus chanted a hymn and sacrificed to such of the gods as they might name.
-- Xenophon, Cyropaedia -- The Education Of Cyrus

We have also Chinese references.

Victor H. Mair provides archaeological and linguistic evidence suggesting that Chinese wū ("shaman; witch, wizard; magician", Old Chinese *myag) was a loanword from Old Persian *maguš "magician; magi". (Mair, Victor H. (1990), "Old Sinitic *Myag, Old Persian Maguš and English Magician", Early China 15: 27–47.)

Mair connects the ancient Bronzeware script for wu 巫 "shaman" (a cross with potents) with a Western heraldic symbol of magicians, the cross potent ☩, which "can hardly be attributable to sheer coincidence or chance independent origination."

This post reaches the core of the history of Classical Antiquity, revealing how the duality within Zoroastrian theology impacted Judaism, as well as Greek and Roman beliefs, leading to Coptic Christianity.

The fire altars of Zoroastrianism illuminate, in my view, the battle between light and darkness which is at the heart of that faith and then appears in all the religions under Persian dominion.

Here is a view, which I do not agree with in totality, but does, I think, address the important subject of how the Persian faith impacted Judasim:

The Subversion of Judaism
In 539 B.C. the great Persian king Cyrus conquered Babylon. His government was a Zoroastrian theocracy. Cyrus had a history of pretending to adopt a religion and then subverting it. In Egypt he claimed to be a god on earth. In Babylon his first act was to worship Marduk, claiming Marduk had sought a righteous prince and Cyrus was he. Later Cyrus mocked Marduk and had his image carted off. Likewise he subverted Baal, worshipping him at first, then appointing Baal’s priests and finally destroying Baal’s monuments and temples. Cyrus repatriated certain grateful Jewish proteges in 532 B.C. The Persian Kings restored them to their land; and designed and helped them build a Zoroastrian-style temple which was completed 516 B.C. after prolonged resistance from native Jews. In 350 B.C. a large number of Jews were exiled from Judea because of opposition to the Persian theocracy. Here was the overwhelming influence of a mighty state religion in a great empire in which the Jews were subjects. Jews were commanded to obey Cyrus in Isaiah 44 and 45. Not only did the Persian kings select the Jewish high priests, Persian Magi even masqueraded as Jewish Priests (Isaiah 66:21).

The only ritual not subverted by the Pharisees was the practice of circumcision. "The Jewish rite did not assume its present form until so late a period as that of the Maccabees (167 B.C.)." Before that time the operation was imperceptible and that may be why the Persians didn’t change it.
-- The Secrets of Zoroastrianism by Mark Willey

Zoroastrianism clearly has had a large influence on Christianity also. There are many aspects of Christianity that were not drawn from Judaism, although that religion was the major predecessor of Christianity. One of the most obvious tenets of Christianity that has its roots in Zoroastrianism is the concept of dualism.

From this basic concept springs many others, many of which are also found in Christianity. Zoroastrianism has a very clear notion of the concepts of heaven and hell. These were the realities that Ahura Mazda and Angra Mainyu had created for themselves.

Zoroastrian tradition holds that, after Zarathustra, there will come three more prophets born of a virgin, each of whom will become pregnant after bathing in a lake which preserves "the seed of the prophet" (Traditional 7). The last of these is to be the Saoshyant, or savior, who will bring about the final judgement. At this time, everyone will be resurrected (this is called Ristakhiz; Traditional 7) and judged a second time by Ahura Mazda, and the final battle between good and evil will take place.

Fresco from the Catacomb of Priscilla
Dating to the mid-third century CE, this fresco is the earliest known image of the magi.