The Catholic Church has always taught that all the baptized share in a unique kingship and priesthood. This does not mean that everyone is a ruler or that everyone gets to preside at Mass.
Instead, it is a different type of priesthood and kingship, one that follows after the example of Jesus Christ, the source of both.
St. Leo the Great explains the qualities of these offices in a sermon featured in the Office of Readings. First he gives some background, explaining how this gives us a "common dignity."
Then St. Leo details what it means to be a "priest" or "king" in baptism.
The type of kingship and priesthood all the baptized inherit is not an earthly one, but a spiritual one. We are "kings" in our spiritual lives, ruling over our passions and directing them in the service of God. We are all spiritual "priests," given the opportunity to sacrifice our very lives to God.
These spiritual offices may not be what we desire in life, but they are given to us through baptism, in imitation of Jesus Christ. If we want to know how to truly be a "king" or "priest," we must gaze upon Jesus on the cross. That is our example.
