Priesthood of the Believer
This (royal) high priesthood of Jesus Christ connects to the "royal priesthood" of believers: "you are a royal priesthood, a holy nation" ( 1 Peter 2:9a ). The obvious reference to Exodus 19:6 suggests that the church functions in this present age as God's New Testament kingdom of priests much like the nation of Israel did in the Old Testament. As such we are responsible to carry out the ministry of proclaiming to the world "the praises of him who called you out of darkness into his wonderful light" ( 1 Peter 2:9b ).
A closely related idea (but without the "royal" connections) is Peter's earlier description of the church as a group of believers who are being (NIV), or should allow themselves to be (NRSV), "built into a spiritual house [Jesus himself being the living and choice cornerstone, 1 Peter 2:4, 6-8] to be a holy priesthood, offering spiritual sacrifices acceptable to God through Jesus Christ" ( 1 Peter 2:5 ). Thus, as fellow priests with Jesus we offer up to God our sacrifices of praise ( Heb 13:15 ), our doing good and sharing ( Heb 13:16 ), and ultimately our present physical bodies in the interest of conforming to his standards ( Rom 12:1-2 ). It is important to observe that here the corporate priesthood of the church shades into the priesthood of the individual believer. Moreover, our ministry in the gospel can be described as an offering of our very life in priestly service to the church ( Php 2:17 ), by which we can produce a harvest of sanctified people whom we present to God as an acceptable offering.
Here's a good Bible study you can download and get a better understanding of the Priesthood of the Believer.