The key problem that priesthood is attempting to solve is the separation between God and humanity which is expressed nicely in the concept of the fall. How do human beings deal with the God who can seem distant yet all powerful; who sets out seemingly arbitrary rules that you cross at your own peril; who rules your life, yet doesn't allow you to make your case? These were the concerns which encouraged ancient societies to identify individuals who would stand between them and divinity - occupying the numinous zone of chaos and fear; bridging the gap and interceding with the all-powerful on behalf of ordinary folk.
If God is distant that heroes are needed who will set out to find him. In ancient mythology the hero must journey through great perils on a quest to acquire something that will set people free. This is a neat metaphor for the journey that priests needed to make to engage with divine power, attempting to gain from God something that will make people's lives more secure and give them hope. Hence priests spent a great deal of time in isolation or in separation from ordinary people as they journeyed inward hoping to make connection with the almighty. They also developed rituals and sacramental actions which were aimed at appeasing divine wrath or making restitution for sins committed (knowingly or unknowingly) against the gods.
This mindset of clerical separation and ritual appeasement is still present in our popular understanding of priesthood. Priests are those who are set aside to deal with God on our behalf. They take time to meditate or pray, developing their own holiness so that they are able to commune with God. They then provide opportunities for ritual actions which enable us to fulfil our own obligations or intercede with God on our behalf.
This ancient and popular view of priesthood is totally blown out of the water by the concept of incarnation. Into our fallen and broken world, the Christ child is born. In this moment, the barrier between God and humanity is brought down because God is now present and is human. We are therefore no-longer able to claim that God is distant or unsympathetic, since in this moment he takes upon himself all the weaknesses and the suffering of ordinary humanity. He lives for many years as a perfectly normal person from first century Nazareth and then steps into the public arena with a mission to demonstrate that the Kingdom of God is already present.
What does the incarnation say about priesthood? It says that if you want to bridge the gap between God and humanity you don't need another human being to act as a intermediary on your behalf, you merely need to look to Jesus. If Christ is present at all times through the Spirit then every word, thought or action is a prayer. You don't need someone else to intercede for you or bridge the gap between you and God, because the gap is only there until you yourself choose to reach out to him. You are your own priest.
The incarnation makes us all priests. We are all called to make our own decision about Christ. No-one else can make it for us. He is either our Lord, or he is not.
There is no barrier between us an God, except the barriers we erect ourselves. We are therefore called to be priests to one another, helping each other to find wholeness and integration. Every act can be sacramental. Every word can bring healing. Every thought can be prayer.
In short, the incarnation has profound implications on priesthood. Into the gap between God and humanity, God himself steps; becoming in Christ the Great High Priest; making us a kingdom of priests who are able to relate to God directly as we seek the good of all.