The Magi's Gifts Symbolize Three Aspects of Christ's Incarnation

In Matthew 2:11 it is written: "and on entering the house they [the wise men] saw the child with Mary his mother. They prostrated themselves and did him homage. Then they opened their treasures and offered him gifts of gold, frankincense, and myrrh." Contrary to popular opinion, the wise men were not kings. They were, rather, according to several Church Fathers, men of intellectual renown and considerable means, most likely from the Orient. Whether such wealth was their own or it was bestowed by royalty, on whose behalf they acted, is the subject of debate.

The wise men's gifts of gold, frankincense and myrrh, seem curious offerings to a child no more than two years of age [and perhaps far younger]. Both popular piety and Church Tradition suggest that the three gifts represent different dimensions or unique offices of Christ Incarnate.

We Three Kings

The verses of the carol "We Three Kings", while not altogether historically accurate, (The wise men were not kings nor were there three of them) are insightful nonetheless: 

Born a King on Bethlehem's plain Gold I bring to crown Him again King forever, ceasing never Over us all to rein

Gold is the gift given to Kings. Jesus Christ is King of the Universe and His Kingdom will have no end.

Frankincense to offer have I Incense owns a Deity nigh Pray'r and praising, all men raising Worship Him, God most high

Frankincense was used in religious ceremonies including the Temple in Jerusalem. It acknowledges Jesus' divinity.

Myrrh is mine, its bitter perfume Breathes of life of gathering gloom Sorrowing, sighing, bleeding, dying Sealed in the stone-cold tomb

Myrrh was the oil with which the dead were anointed prior to internment. It prefigures the sacrificial death of Christ as the Lamb of God.

The three offerings may also pay homage to the three divine offices of Jesus' Ministry as Priest, Prophet and King. It is worth noting that Jesus, a small child, had little use for such extravagant gifts. It is Mary who receives them and presumably keeps them on behalf of her Son. Catholic Tradition has long understood Mary to be the "Ark of the New Covenant" in whom God Incarnate literally resided. [The gifts also correspond to the contents of the Ark of the Covenant as listed in Hebrews 9.]

Here again, gold symbolizes Christ's kingship, frankincense symbolizes His priesthood and myrrh stands for His prophecy.

To some, the nature of the gifts is significant. Gold is the only gift of the three that endures. The other two, frankincense and myrrh, are spices that are temporary in nature [representing Christ’s life and His atoning death]. Gold [His kingdom] is eternal in nature.

There are numerous iterations on the aforementioned themes including one school of thought that sees the gifts as honorary but not symbolic.