So, let's begin at the beginning. First, this account is only found in one of the four gospel accounts: Matthew, Chapter 2. Second, we have to discriminate between what is told in Scripture and what has attached itself to the story over time, in the culture of Christianity.
We know men came from the East, following a star, to worship the new-born baby Jesus. We know that they brought with them gifts of gold, frankincense, and myrrh. What we do not know is how many in number they were. Scripture is simply silent on the matter. That they were three in number is attached to the number, or perhaps stated better, the kind of gifts brought: gold, frankincense, and myrrh. One might naturally conclude there were then three "wise men", but we simply cannot say with certainty. However, it does make for good lyrics in a well-known song 😉.
Concerning the identity of these men, Anglican Bishop N.T. Wright observes the following in his commentary, Matthew for Everyone:
"The arrival of the 'Magi' (that's the word Matthew uses for them; it can refer to 'magicians', or 'astrologers', or experts in interpreting dreams, portents and other strange happenings)...."
Further, Bishop Wright offers this concerning the gifts themselves:
"The gifts that the Magi brought were the sort of things that people in the ancient world would think of as appropriate presents to bring to kings, or even gods."
While much more can be said, what is most useful to observe for purposes of this article is that these men were "wise men" but not kings themselves. Further, they were wise enough to recognize that something of supreme earthly value, and even perhaps divine value, was happening through the birth of Jesus. Of course, we know in hind-sight that both are true, which brings us to the gifts themselves and their symbolism. Here, I will allow the well-known and respected commentary of William Barclay, The Gospel of Matthew, Vol. 1, speak to these things:
"Gold is for a king, frankincense for a priest, myrrh for one who was to die - these were the gifts of the wise men, and, even at the cradle of Christ, they foretold that he was to be the true king, the perfect high priest, and in the end, the supreme Savior of the world."
A final note: this, of course, is the great twist of the plot. Gold is appropriate to a king. Incense is appropriate to a priest. However, we have to ask the question: if a savior comes to save, how can he save anyone, if he's dead? The answer, of course, is that Jesus came to die. As Jesus himself teaches us in John 10:18, "No one takes [my life] from me, but I lay it down of my own accord. I have authority to lay it down and authority to take it up again. This command I received from my Father." And, in so doing, Jesus would destroy the power of death by his own death, because he rose to life again on the 3rd day!