In the Lord's Prayer, "Thy kingdom come" is a request that the reign of God be fully established. Most Christians believe that this is going to occur when Jesus returns to the earth to establish his rule over all humankind.
Jesus' instructions to his disciples on prayer are part of the Sermon on the Mount. At this time in Jesus' ministry, he was proclaiming the kingdom of God that had been preached by the Old Testament prophets, according to which God's promised Messiah would restore the kingdom of Israel and rule as David's successor. The prophets identified the Messiah as both a divine redeemer and as a descendant of David. Some Christians believe that the Jewish rejection of Jesus as the promised Messiah-King resulted in the delay of the setting up of the kingdom until Jesus' return, at which point the Jews are expected to accept him as their king and live in obedience to and at peace with their God.
In asking for the kingdom to come, Christians commit themselves to living as if that kingdom were already in place by obeying God and praying for the needs that they see around them. They also believe that they are carrying out the mission of the kingdom by spreading the gospel and making converts.