Magi: probably astrologers (who are also astronomers) and priests from Persia. It is striking that heathens with clearly false views about the heavens would figure prominently in the story of Jesus. Perhaps the point is that if such as these could recognize who Jesus was, surely the Jews should have.
We don't know whether the Magi gave up their un-Christian practices and beliefs as a result of this encounter. This gives credence to the idea that the key to redemption is in recognizing Jesus as Messiah and worshipping him, not in adhering to correct church doctrine. Certainly there are many throughout church history with incorrect doctrine; in fact, you could make the case that no one (except for me) has correct doctrine in every topic. As long as the basic facts of Jesus as Messiah are recognized, God allows for our errors in non-essentials.
From the east: The Magi likely lived in Persia perhaps 400 or 500 miles away (the reverse route of Abraham: from Jerusalem; first north, then east, then southeast).
The Magi may have been priests of the monotheistic Zoroastrian religion. The Magi had influence in the court much as Daniel did; perhaps Daniel acted as an early Magi. This religion had the concept of a savior. Perhaps the Magi thought Jesus was this savior.
King of the Jews: The Jews believe the Messiah will be a king. In fact, one of the reasons they reject Jesus as being the Messiah is because he was not a king! A few Old Testament passages indicating the Messiah would be king...
A Psalm for Solomon. Give the king thy judgments, O God, and thy righteousness unto the king's son. The kings of Tarshish and of the isles shall bring presents: the kings of Sheba and Seba shall offer gifts. (Psalms 72:1,10)
Rejoice greatly, O daughter of Zion; shout, O daughter of Jerusalem: behold, thy King cometh unto thee: he is just, and having salvation; lowly, and riding upon an ass [donkey], and upon a colt the foal of an ass [donkey]. (Zechariah 9:9)
I saw in the night visions, and, behold, one like the Son of man came with the clouds of heaven, and came to the Ancient of days, and they brought him near before him. And there was given him dominion, and glory, and a kingdom, that all people, nations, and languages, should serve him: his dominion is an everlasting dominion, which shall not pass away, and his kingdom that which shall not be destroyed. (Daniel 7:13,14)
Star in the east: Jerusalem was to the west but the star was in the east. This implies this star was an astronomical event, not a supernatural star guiding their journey. They went to Jerusalem because they knew from the Old Testament that is where the Messiah would be, not because the star pointed the way.
In 5 B.C. there was a very bright comet which appeared first in the east, then shifting position in the sky slowly southward for 70 days. When the Magi arrived in Jerusalem, the comet was in the sky above the direction of Bethlehem to the south. This comet "pointed" downwards, with the tail upwards, like an arrow pointing towards the town of Bethlehem.
For the previous two years there had been unusual and important astronomical events.
From the date of this comet, we should assume Jesus was born in 5 B.C.
Probably the Magi travelled slowly since they would want to spend time studying the comet. This was an unusual astronomical event.
Worship: The king of the Jews was to be worshipped and it was worth spending perhaps 6 months of your life to do so. Notice that the Magi made a pilgrimage in their worship of Jesus.
Herod was jealous of Jesus, the king. This jealousy led to the slaughter of the innocents: the children 2 years old and under. Probably the Magi mentioned the astronomical events over the past two years; their chattiness with Herod the tyrant cost these children their lives. We should wonder why John the Baptist was not also killed at this time; perhaps Joseph warned Zacharias.
Herod assumed that the Jewish leaders would know something about the Messiah, in particular, where he was to be born. This fact was important to Herod because he planned to execute him. It is ironic that decades later it was these same Jewish leaders who executed Jesus even though Pilate wanted to release him.
The Jewish leaders knew the Messiah was to be from Bethlehem. But later, they don't know that Jesus was born there (and they don't even bother to ask — they don't want to know the truth.)
Others said, This is the Christ. But some said, Shall Christ come out of Galilee? Hath not the scripture said, That Christ cometh of the seed of David, and out of the town of Bethlehem, where David was? They answered and said unto him, Art thou also of Galilee? Search, and look: for out of Galilee ariseth no prophet. (John 7:41,42,52)
This is a quote from Micah...
But thou, Bethlehem Ephratah, though thou be little among the thousands of Judah, yet out of thee shall he come forth unto me that is to be ruler in Israel; whose goings forth have been from of old, from everlasting. (Micah 5:2)
It seems odd this passage would inform us that Herod asked when the star appeared but not reveal to us the answer. The Magi seem to be rather naive in dealing with Herod; after leading Herod to the young child Jesus they head back to Persia.
Herod is a liar. He has no intention of worshipping Jesus.
The star which they saw in the east now guided them to the precise location of Jesus in Bethlehem. This second star could not be merely an astronomical conjunction of planets; it was probably an apparition which guided the way. This apparition probably looked like the astronomical conjunction of planets that had led them to travel to Jerusalem, that is why verse 9 refers to it as "the star which they saw in the east".
The star guided them to the very house in which Jesus was.
They rejoiced when the saw the star. After a long 3 month journey they were probably quite relieved to find that their quest was not in vain. It seems the heavens were confirming that the prophecies concerning the birth of the Messiah that led them to Jerusalem were true.
And when they were come into the house, they saw the young child with Mary his mother, and fell down, and worshipped him: and when they had opened their treasures, they presented unto him gifts; gold, and frankincense, and myrrh. (Matthew 2:11)
It is a Catholic tradition to worship the child Jesus but Protestants reject this idea. They seem to think that Jesus wasn't deity until he became an adult.
Part of their worship involved giving material gifts of great worth. These likely paid for the care of Jesus for many years, especially after Joseph died.
Jesus is no longer in the manger. Likely he has already been presented by Mary in the temple. He is perhaps as old as 2 years old since Herod killed all the children up to that age based on his diligent inquiry into the time the Magi first saw the star...
Then Herod, when he saw that he was mocked of the wise men, was exceeding wroth, and sent forth, and slew all the children that were in Bethlehem, and in all the coasts thereof, from two years old and under, according to the time which he had diligently inquired of the wise men. (Matthew 2:16)
This is a strange trip of the Magi. They unwittingly guide Herod to Jesus so he can kill him. They are rather dull-witted. They made no effort to try to save Jesus from the danger they placed him in; they merely went back to Persia self-satisfied that they had worshipped him, oblivious to the fact that he would have to be uprooted and taken to Egypt for years for safety. With friends like these who needs enemies.
After the Babylonian exile the Jews continued to live in Persia. The Magi were likely familiar with Old Testament prophecies regarding the time of the coming of the Messiah. They made a months-long journey to worship this Messiah as a young child. It seems odd that these Persian astrologers could recognize the coming of the Messiah but that the Jewish leaders missed it. They weren't paying attention.
Daniel's prophecy gives the date of the coming of the Messiah...
Seventy weeks are determined upon thy people and upon thy holy city, to finish the transgression, and to make an end of sins, and to make reconciliation for iniquity, and to bring in everlasting righteousness, and to seal up the vision and prophecy, and to anoint the most Holy. (Daniel 9:24)
Know therefore and understand, that from the going forth of the commandment to restore and to build Jerusalem unto the Messiah the Prince shall be seven weeks, and threescore and two weeks. (Daniel 9:25)