What does it mean to be a part of the royal priesthood? For starters, it means Jesus is our king, and we are united under his lordship. Our royal identity derives from his divine monarchy.
Romans 5.17 says: 
For the sin of this one man, Adam, caused death to rule over many. But even greater is God’s wonderful grace and his gift of righteousness, for all who receive it will live in triumph over sin and death through this one man, Jesus Christ.
Biblical kingship means three things:
                        and service
In Genesis 1, Adam and Eve were given dominion over the Garden of Eden, which meant they were entrusted with the conservation and development of God’s creation. Their task was not to dominate, but to nurture life and perpetuate a life-giving ecosystem. They were caretakers, invested in the success of project Earth and cooperating with God to see it fulfilled. 
Justice means making and keeping things right, just as God wants them to be. There are two kinds of justice in our modern understanding: retributive and restorative justice. Retributive justice involves punishment for the purpose of dissuasion. For example, sending a thief to jail to punish him for his theft, or sending a disobedient child to his room for his poor behavior. The New Testament, however, guides us to restorative justice. For example, Galatians 6.1-2 says:
If another believer is overcome by some sin, you who are godly should gently and humbly help that person back onto the right path. And be careful not to fall into the same temptation yourself. Share each other’s burdens, and in this way obey the law of Christ. 
Restorative justice means our actions in response to sin are designed not to punish, but to make things right. For example, a thief might be forced to return what he has stolen, repay what he has cost the victim, and offer his life in service to them for a set amount of time. Restorative justice in the means by which we get from the distortion of sin back to the blessing of God. However, the more heinous the crime, the more complex the restoration. Crimes of magnitude affect hundreds of people, leaving us with little confidence that a restorative solution can be found. Thus, the biblical concept of restorative justice essentially begs for God’s involvement, because we are incapable of providing sufficient recompense through our own cleverness, resources, or counsel.
We need God’s help to make the world just.
Finally, biblical kingship means service. Ultimately, only Jesus is Lord, and we all follow him. Kings serve The King. Throughout the stories of kings in the historical books of our Bible, kings are adjudicated based on whether or not they follow the way of the Lord.
As goes the king, so go the people.