Question: "What does it mean that Israel is supposed to be "a kingdom of priests and a holy nation" (Exodus 19:6)?"
Answer: In Exodus 19 God announces for Israel a conditional covenant. If they would keep His covenant, Israel would be a people for His own possession and a kingdom of priests and a holy nation (Exodus 19:5–6). The people of Israel responded that they would fulfill what God would command them, and thus they sealed the covenant with a commitment (Exodus 19:8). What follows in Exodus includes six hundred and thirteen commandments, including the Ten Commandments of Exodus 20.
Earlier, God had made a covenant with Abraham (Genesis 15—17), and He committed to keeping that covenant through the line of Isaac and Jacob (see Genesis 50:24 and Exodus 2:24). God had committed to making the nation of Israel a mighty nation with a land (Genesis 15:18–21), a king (2 Samuel 7:10–16), and an eternally blessed people (Jeremiah 31:31–37). Those covenants were all unconditional—God had obligated Himself without any requirement on the part of the recipients of His covenant promises.
The covenant God made with Israel through Moses was different, in that it was conditional. It came with an if. It required Israel’s obedience in order to receive the conditions of blessing, and that first condition of blessing God revealed was that Israel would belong uniquely to Him and would be a kingdom of priests and a holy nation. This covenant (often called the Mosaic Covenant because it was made through Moses) was also unique in that it was a teaching tool to lead people to Christ (Galatians 3:24). It was never a means of salvation, because salvation is never by works of law (Romans 3:20) but rather by grace through faith (Ephesians 2:8–9).
Israel broke the covenant (Jeremiah 31:32) and forfeited the blessings offered therein. However, Paul explains that Abraham was the father of three different kinds of descendants in Romans 4. He is the father of many according to the flesh (Israel in general, Romans 4:1), he is the father of those who believe who are not of Israel (Gentile believers, Romans 4:11), and he is the father of all who believe and are of the nation of Israel (Jewish believers, Romans 4:12). While the nation of Israel broke the conditional “old” covenant in unbelief, those of Israel who would believe in Jesus (the third group of Abraham’s descendants) are called out using similar terminology. Peter, writing to Jewish Christians scattered throughout the Roman world, calls those Jewish believers “a chosen race, a royal priesthood, a people for God’s own possession, so that you may proclaim the excellencies of Him who has called you out of darkness into His marvelous light” (1 Peter 2:9, NASB).
In Revelation 1:6 John asserts that the church is a kingdom and priests to God the Father. This would include both believing Jews and believing Gentiles—the second and third kinds of descendants of Abraham described in Romans 4:11–12. Thus, the only ones of Abraham’s descendants who were not a kingdom and priests were those who broke the (Mosaic) covenant and were not believing.
These who believe are referred to as a kingdom, as those who believe in Jesus are transferred to His kingdom (Colossians 1:13). Believers are also considered priests in that we are interceding on behalf of people that they might believe in the Lord. Those who are called a kingdom and priests have an obligation to walk in a manner worthy of His calling (Ephesians 4:1) and ought to be actively proclaiming His excellencies (1 Peter 2:9) so that others may come to know Him.