The phrase "royal priesthood" means that every believer in God has a special role and responsibility, just like priests in the Bible. It signifies that we have direct access to God and are called to represent Him in our daily lives. Full definition
«Only by praying together with their children can a father and mother — exercising their royal priesthood — penetrate the innermost depths of their children's hearts and leave an impression that the future events in their lives will not be able to efface.» (faith.org.uk)
Yet this is our claim: that we become members of Christ, putting on Christ, participating in Christ's royal priesthood. (religion-online.org)
«You are a chosen race, a royal priesthood, a holy nation, God's own people, that you may declare the wonderful deeds of God who called you out of darkness into marvelous light». (religion-online.org)