They have been addressed as “wise”, “kings”, or simply “magi”, and in the book of Matthew he tells us about the three wise men who went on a pilgrimage seeking Jesus, following a rising star to Jerusalem.
Matthew referred to them as magi who were wise in understanding the stars and their relations to events on Earth. They were also from "the East" where astronomy had been practiced as early as the 2nd millennium BC.
King Herod was the King of the Jews from around 36 B.C.-1 A.D. and when the wise men from the East arrived they asked Herod and the people where “the one having been born King of the Jews” was because they had seen his star in the East and wished to pay homage.
In Ezekial 43:2 it discusses this topic of the birth of Jesus coming from the East, as did the star.
2 And behold, the glory of the God of Israel was coming from the east.
And the sound of his coming was like the sound of many waters,
and the earth shone with his glory.
Hearing this, King Herod became very suspicious as he did not want someone to usurp his throne. In response, Herod requested information, so he called upon his chief priests and scribes and wanted to know where this “one having been born of the King of the Jews” was and in response they told him in Bethlehem, citing Micah 5:2 in the English Standard Version (ESV)
2 But you, O Bethlehem Ephrathah,
who are too little to be among the clans of Judah,
from you shall come forth for me
one who is to be ruler in Israel,
those coming forth is from of old,
from ancient days.
It was after hearing this that King Herod secretly summoned the three wise men to ask when the star had appeared, as a way to find out the age of the child. This way, Herod said, he could go and worship him as well.
The wise men set out on a journey to Bethlehem in search of the child. When they arrived in Bethlehem the star they followed was stopped over the place where the child was (Matthew 2:9) and they were overjoyed.
Often regarded as the oldest one the three and described as having a long beard, white hair and just generally being old, Saint Melchior comes from Persia and brought a gift of gold to Jesus.
The gift of gold is thought to be symbolic of Jesus' kingship over the world. As said in Daniel 7:13-14, of the English Standard Version (ESV).
13 I saw in the night visions,
and behold, with the clouds of heaven
there came one like a son of man,
and he came to the Ancient of Days
and was presented before him.
14 And to him was given dominion
and glory and a kingdom,
that all peoples, nations, and languages
should serve him;
this dominion is an everlasting dominion,
which shall not pass away,
and his kingdom one
that shall not be destroyed.
Saint Caspar is described as having a reddish beard, younger than Saint Melchior and older than Saint Balthasar and presents Jesus with frankincense, from an assistant. Saint Caspar is often depicted kneeling at the foot of Jesus.
The symbolization of frankincense is thought to be the symbol of deity, or holiness and righteousness.
This can be seen in the Bible in the passage of Colossians 2:9-10:
9 For in him the whole fullness of deity dwells bodily, 10 and you have been filled in him, who is the head of all rule and authority.
Saint Balthasar was the youngest of all three and believed to be as described by Saint Bede in the 8th century, "of black complexion, with a heavy beard" and with this he goes on to describe Saint Balthasar's gift of "myrrh he held in his hands prefigured the death of the Son of man" as both gift and description of symbolization.
It is believed that the gift of myrrh was symbolizing the death of Jesus as it was commonly used to embalm bodies. We can see this in a passage in the Gospel of John in 19:38-40.
38 After these things Joseph of Arimathea, who was a disciple of Jesus, but secretly for fear of the Jews, asked Pilate that he might take away the body of Jesus, and Pilate gave him permission. So he came and took away his body. 39 Nicodemus also, who earlier had come to Jesus by night, came bringing a mixture of myrrh and aloes, about seventy-five pounds in weight. 40 So they took the body of Jesus and bound it in linen cloths with the spices, as is the burial custom of the Jews.
The wise men were divinely led to this place and their departure would be as well, after visiting with the Jesus the three wise men had a dream, God warned them not to return to Herod. The magi left Judea and returned to their own countries each of a different route.
Joseph had a dream about Herod coming for the child as well and with that Joseph and his family all fled to Egypt to escape King Herod. Joseph, Mary, and their son stayed in Egypt, eventually moving to Nazareth to avoid Archelaus, Herod's son.
Upon King Herod having heard of being outwitted when the wise men did not return, the "Massacre of the Innocents" would occur, in which all boys up to two years old would be killed in the vicinity of Bethlehem, unsuccessful for Herod and a devastating tragedy.
The Three Wise Men would end up meeting again later in life around 54 A.D. in the Kingdom of Armendia, avoiding Herod their whole life to celebrate Christmas together one last time.
The magi are regarded as saints in Western Christianity and their feast day is January 6th, the Feast of Epiphany.