Updated: May 19, 2020
Suppose someone asked you what understand under the idea of the "Kingdom of God" (Kingdom of Heaven). What would your response be? The easy answer would be something along the lines of "Everywhere God reigns, in heaven and on Earth". This much is true, but in the New Testament there is explicit references to a new Kingdom of God that is "at hand".
In those days John the Baptist came preaching in the wilderness of Judea, “Repent, for the kingdom of heaven is at hand.”
Matthew 3: 1-2, ESV
So what is this this Kingdom of God that John preached is "at hand?". Surely it cannot be synonymous with God's control over His creation, since that is not something that is "at hand" but something that was active since Genesis chapter 1.
At the heart of this theme of the Kingdom of God is the idea of God’s messianic kingdom. It is a kingdom that will be ruled by God’s appointed Messiah (that is Jesus Christ), who will be not just the Redeemer of His people, but their King. So when John speaks of the radical nearness of this breakthrough, the intrusion of the kingdom of God, he’s speaking of this kingdom of the Messiah. To fully understand the meaning of the Kingdom of God, we need to go to the beginning... 
In the Beginning
In the beginning God created the heavens and the earth. He created the sun and the moon, the stars the plants the animals and finally the crown of His creation: humanity. God (Yahweh) is the Creator-King who effortlessly creates everything out of nothing in six days to be “very good”.
And God saw everything that he had made, and behold, it was very good. And there was evening and there was morning, the sixth day.
Genesis 1:31, ESV
Since God is the creator, He has everything under His authority and hence is able to create human beings and pass dominion over creation to humanity, to allow us to rule in God's place over creation.
Then God said, “Let us make man in our image, after our likeness. And let them have dominion over the fish of the sea and over the birds of the heavens and over the livestock and over all the earth and over every creeping thing that creeps on the earth.”
Genesis 1:26, ESV
From Genesis chapter 1 we can clearly begin to see the image of a certain type of Kingdom. God is the ruler over the Kingdom (creation) but we have been given dominion. Since the beginning, the creation was meant to be God reigning through a vice-king who was to submit to His will and in turn, make creation subject to it as well. Adam was made in God’s image to reflect His wisdom and righteousness. This idea is again re-enforced in Psalm 115:
The heavens are the Lord's heavens, but the earth he has given to the children of man.
Psalm 115:16, ESV
One of the commands given to humanity at our creation was that we ought to 'keep' the garden in Genesis 2:15. The Hebrew word translated as keep is 'samar' which means 'to keep, guard, keep watch and ward, protect, save life'.
Adam, therefore, was supposed to recognise the lies of Satan and 'guard' the garden against the lies of Satan. But rather than exercising dominion over the serpent, Adam transgressed and instead the serpent became his ruler. Instead of ruling as God's vice-king, Adam and Eve rebelled against God and their sin was counted toward the rest of humanity by virtue of Adam being our federal head.
Therefore, just as sin came into the world through one man, and death through sin, and so death spread to all men.
Romans 5:12, ESV
Kingdom Lost, Salvation Promised
Right after the fall in the sad state humanity was in God made a promise:
I will put enmity between you and the woman, and between your offspring and her offspring; he shall bruise your head, and you shall bruise his heel.
Genesis 3:15, ESV
Hence even though Adam rebelled, God promised that He will provide salvation from sin's new dominion over humanity. The kingdom we were in before the fall is lost, now we look forward to a new Kingdom, one that will be ruled by a promised King that will crush the head of the serpent and free us from our sin - the Kingdom of God. The new rule of sin is immediately portrayed in Genesis 4. Just before Cain murdered his brother Abel, The LORD spoke to him:
The LORD said to Cain, “Why are you angry, and why has your face fallen? If you do well, will you not be accepted And if you do not do well, sin is crouching at the door.Its desire is contrary to you, but you must rule over it.”
Genesis 4:6-7, ESV
We ought to rule over sin, but Cain's failure shows that it's actually sin ruling over us. The writer of Hebrews clearly states that if it were not for the death of Christ, we would have been subjected to a life of lifelong slavery.
Throughout the rest of the Old Testament the messianic King is promised and a lot of the Old Testament figures point toward the coming of this King. A simple example would be the story of King David, who fought on behalf of the Israelites against Goliath. David, the shepherd-king fighting on behalf his nation (Kingdom) to defeat Goliath who is clothed in scales (think serpent) and crushes his head (Genesis 3) to deliver his people from the Philistines (sin).
In the Kingdom of God (where men are no longer slaves to sin) the Messiah was to be the head. He is everywhere in the Old Testament set forth as a king. (See Genesis 49:10; Numbers 24:17; 2 Samuel 7:16; Isaiah 9:6, 7; 11.; 52.; 53.; Micah 4.; and Psalms 2.; 45.; 72.; and 110.) .
It is also of great importance to note the specific promise made to David by God, telling him that one of his descendants will always sit on the throne, reigning over God's people.
And your house and your kingdom shall be made sure forever before me. Your throne shall be established forever.’”
2 Samuel 7:16, ESV
The Kingdom of God is at Hand
So what did John mean when he was preaching that the Kingdom of God is at hand? Quite simply, that the King will come very soon to deliver his people from their bondage to sin! When He came, Jesus inaugurated God’s kingdom. He didn’t consummate it (meaning complete it fully), but He started it. On the cross He took on the wrath reserved for us because of our transgressions of God and broke the back of sin's rule over us.
We know that our old self was crucified with him in order that the body of sin might be brought to nothing, so that we would no longer be enslaved to sin. For one who has died has been set free from sin. Now if we have died with Christ, we believe that we will also live with him. We know that Christ, being raised from the dead, will never die again; death no longer has dominion over him. For the death he died he died to sin, once for all, but the life he lives he lives to God. So you also must consider yourselves dead to sin and alive to God in Christ Jesus.
Let not sin therefore reign in your mortal body, to make you obey its passions. Do not present your members to sin as instruments for unrighteousness, but present yourselves to God as those who have been brought from death to life, and your members to God as instruments for righteousness. For sin will have no dominion over you, since you are not under law but under grace.
Romans 6:6-14, ESV
And when He ascended into heaven, He went there for His coronation as the King of kings and Lord of lords. This Kingdom of God belongs to Christ, not as the Logos, but as the Son of Man, God manifest in the flesh: which fulfills God's promise to David that his throne shall be established forever, as Jesus comes from the tribe of Judah and is a direct descendant of David. 
We know that after the crucifixion, after the resurrection that Jesus (the God-man) was given all authority in heaven and on Earth, and hence His reign has begun:
And Jesus came and said to them,“All authority in heaven and on earth has been given to me.Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, teaching them to observe all that I have commanded you. And behold,I am with you always, to the end of the age.”
Matthew 28:18-20, ESV
Your Kingdom Come
This is part of the Lord’s Prayer. In this specific verse, the Lord’s Kingdom is connected to His will, so that our kingdom obedience means surrendering our lives to do the King’s will. It also means that God's church will expand as more people come to faith in Jesus Christ, being elected by the Father, drawn and regenerated by the Holy Spirit such that they are no longer slaves to sin, but slaves to Christ. This is also the answer that the Heidelberg Catechism supplies:
What is the second petition?
So rule us by your Word and Spirit that more and more we submit to you. Preserve and increase your church. Destroy the works of the devil, every power that raises itself against you, and every conspiracy against your holy Word. Do all this until the fullness of your kingdom comes, wherein you shall be all in all.
LORD's Day 48, Question 123