ANSWER: The phrase 'kingdom of God' is used at least seventy times in the King James New Testament. The gospel of Mark uses it at least fifteen times (Mark 1:14 - 15, 4:11, etc.) while it is recorded in the book of Luke thirty-three times (Luke 4:43, 6:20, 7:28, etc.)! The synonymous phrase "kingdom of heaven" is recorded an additional thirty-three times in the KJV translation. Interestingly, all thirty-three occurrences are found solely in the gospel of Matthew (e.g. Matthew 3:2, 4:17, 5:3, etc.).
A short answer to your question is that the Kingdom, at the present time, is the rule and governance of the Godhead (the Father and Jesus Christ) over all things. The Bible speaks of it existing in the past (Daniel 4:17, 25, 34, 5:21), present (Matthew 12:28, Luke 17:20 - 21, Colossians 1:13), and future (Daniel 2:44 - 45, 7:13 - 14, Revelation 11:15 - 18, 20:4 - 15, 21:1 - 22:5). Right now, although it rules the entire universe, it is not actively governing over the affairs of man. God is, nevertheless, ruling in the lives of all those who have his Holy Spirit and are obedient to him.
When Jesus, who represents the kingdom, returns to earth a second time, all true Christians (whether they are in the grave or still alive) will be changed into spirit beings. This transformation, known as the first resurrection, is necessary for the saints to enter into salvation and to rule on earth (1Corinthians 15:50 - 53, see also 1Thessalonians 4:13 - 18, John 3:3 - 7). Christ, along with his resurrected and immortal saints, will set up the kingdom on earth (Revelation 5:10) and rule for 1,000 years (the Millennium) from the planet's capital city of Jerusalem. Their reign shall be glorious and, according to the prophet Daniel, their dominion will never end (Daniel 2:44).
After a thousand years are completed, all those who never got a full chance at salvation will be resurrected to receive their opportunity to repent, live the way of righteousness, and enter into the Kingdom. The Father will then bring the New Jerusalem from heaven to the new earth and His government will rule the entire universe from this glorious city for all eternity. The earth will literally become the center of all creation FOREVER.
Those who will be allowed into the Eternal's reign of righteousness must receive it like a little child (Mark 10:15), be doers of his will (Matthew 7:21 - 23) and keep his Commandments (Matthew 19:16 - 17; Revelation 22:14). They must be willing to seek it (Luke 12:31), to give up all things in order to possess it (Luke 18:29 - 30), love their fellow man (Matthew 25:31 - 46) and be willing to suffer persecution for its sake (Matthew 5:10).
Who will NOT be allowed to enter the Kingdom of God? According to the Bible, those who will not be welcomed into eternity are those who are adulterers, who practice sorcery, are contentious, have outbursts of anger, are selfish, those who promote false doctrines, those who cause divisions among believers, murderers and more (Galatians 5:19 - 21). The Apostle Paul also states in the book of Ephesians that fornicators and those who are covetous will not enter it either (Ephesians 5:5). Toward the end of the book of Revelation, we are also told that those who are cowards, perverts, traitors, those who are immoral, those who murder, those who practice magic and worship idols, and all liars will not be in the Kingdom. What they will be in, however, is the lake of fire (Revelation 21:8) created expressly for carrying out an eternal death penalty on all those who willfully reject salvation.