What is the "Kingdom of God" supposed to be? everlasting blissful happiness and peace on earth
heaven after death / afterlife
a metaphorical way to describe that righteousness would be made on earth by the power of the holy spirit, and that Jesus' parables give a sample and foreshadow of what the real Kingdom of God on earth will be like; in order to achieve the Kingdom of God, one must obey his commandments as well as continue the covenants given to Israel before Jesus.
The Kingdom of God or Heaven literally means the Realm where God dwells.

The Kingdom of God may be divided into two types based on it's establishment.

 1. At the First coming of Jesus Christ

When Jesus came to earth as a humble man to suffer and die for us, He's goal was to bring down the Kingdom of God, which was also His role as the promised Messiah. But at His first coming, the Kingdom of God was not to be established physically on earth, which was the expectation of the Jews. Jesus came to establish the Kingdom of God in the hearts of men. Jesus was preparing the people for the coming kingdom, which was to be established soon after He finished His work on the cross. John the Baptist was proclaiming that the Kingdom of God was near.


  Matthew 3:1-2 (NIV) In those days John the Baptist came, preaching in
  the wilderness of Judea 2 and saying, “Repent, for the kingdom of
  heaven has come near.”


Jesus started explaining to the people what the Kingdom of God is. He used metaphors like  the sower, mustard seed, hidden treasure, weeds, yeast and fishing net (see Matthew 13). But until it was established from the day of Pentecost, which was the beginning of the Church era, it was not fully understood. Therefore, this Church era is the first establishment of Kingdom of God and it's dominion is in the hearts of mankind.

For instance,


  Matthew 13:23 (NIV) But the seed falling on good soil refers to
  someone who hears the word and understands it. This is the one who
  produces a crop, yielding a hundred, sixty or thirty times what was
  sown.”


Here, Jesus said that the Kingdom of God can start from a single person, which can yield many other believers. Peter was the first man to preach the Gospel on the Day of Pentecost and 3000 people were baptized that day. And the apostles went on to preach everywhere and the number of Christians kept on growing and growing. 

2. At the Second coming of Jesus Christ

Jesus will come again with all His glory and power to establish His kingdom physically on earth forever. This present world is in the hands of Satan but when Jesus comes, Satan will be defeated and the eternal physical Kingdom of God will be established on earth forever and ever. Amen.


  Revelation 19:11-20 (NIV)
  - excerpt
  
  I saw heaven standing open and there before me was a white horse,
  whose rider is called Faithful and True. With justice he judges and
  wages war. He is dressed in a robe dipped in blood, and his name is the Word of
  God. The armies of heaven were following him, riding on white horses
  and dressed in fine linen, white and clean. Coming out of his mouth is
  a sharp sword with which to strike down the nations. “He will rule
  them with an iron scepter.” On his robe and on his thigh he has this
  name written: king of kings and lord of lords.
  Then I saw the beast and the kings of the earth and their armies
  gathered together to wage war against the rider on the horse and his
  army. But the beast was captured, and with it the false prophet who
  had performed the signs on its behalf. 
  The two of them were thrown alive into the fiery lake of burning
  sulfur.