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 Matthew.
The short answer to your question is that the Kingdom is the rule and governance of God (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).
Although God's government rules the entire universe, it is not actively governing the affairs of man. The Almighty is, nevertheless, ruling in the lives of all those who have his Holy Spirit.
When Jesus returns to earth a second time, all true Christians will be changed into spirit beings. This transformation, known as the first resurrection, is necessary for the saints to rule on earth (1Corinthians 15:50 - 53, 1Thessalonians 4:13 - 18, John 3:3 - 7).
Christ, along with his immortal saints, will set up God's rule on the earth (Revelation 5:10) and rule for 1,000 years (the Millennium). Their reign shall be glorious and 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 eternal life. The Father will then bring his newly created New Jerusalem to the new earth and His government will rule the universe from this glorious city. The earth will literally become the center of all creation.
Those who will be allowed into the Eternal's kingdom 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) and love their fellow man (Matthew 25:31 - 46). They must also be willing to suffer and possible die for its sake (Matthew 5:10).
Who will not be there?
Who will not be allowed to enter the Kingdom? According to the Bible, those who will not be welcomed into eternity are those who are adulterers, are contentious, who practice sorcery, 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 further defines the Kingdom by stating that fornicators and those who are covetous will not enter it (Ephesians 5:5).
Toward the end of Revelation, we are also told that those who are cowards, perverts, traitors, immoral, murderers, practitioners of black magic and worship idols, and all liars will not be in the Kingdom of God. 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.