The dream.
The king's dream is described in Dan 2: 29 - 36.
Briefly, the king had gone to bed wondering what would happen the his kingdom after he died, and the dream was to show him what would happen.
He dreamt about a large statue, looking like a man. This statue had a head of gold, chest and arms of silver, belly and thighs of bronze, legs of iron, and feet partly of iron and partly of clay. Then a large rock, cut out without hands, smashed into the feet of the statue and shattered it. The wind blew away the remains of the statue, while the rock became a huge mountain that filled the whole Earth.
Quite a dream. No wonder it woke him up and worried him!
Daniel gives the interpretation of the dream in the rest of the chapter.
In vs 36 - 38 he states quite clearly that Nebuchadnezzar is the head of gold. This does aptly represent the neo-Bablyonian empire, which was one of the largest and most affulent empires of ancient times. The fabled hanging gardens, one of the seven wonders of the world, built by Nebuchadnezzar for his wife, were just one of the magnificent edifices gracing the city.
In vs 39, Daniel goes on to say that after Nebuchadnezzar, another kingdom would arise that would be inferior to Bablyon. What kingdom was this? The Bible give us the answer in Daniel 5:28, where it says that the Bablyonian empire was given to the Medo-Persian empire. History tells us that the city of Bablyon fell to the armies of Media and Persia in October 539 BC. The Medo-Persian empire then took over as the major world power. The Medes and the Persians were related tribes from what is not Iran who joined forces to carve out a world empire. Initally, the Medes were the dominant tribe but eventually the Persians became dominent. History also tells us that although the Medo-Persian empire occupied a larger area than the Bablyonian, it never reached the heights of luxury or learning that the Bablyonian did. So the prophecy was fulfilled.
Daniel continued in vs 39 to say that a third kingdom would arise and rule the world. Who was this? Again the answer is given in the Bible as well as history. In Daniel 11:2 it says that Greece would overthrow the Medo-Persian empire. This is confirmed by history, when Alexander the Great conquored the then-known world, overthrowing the Persians in 331 BC.
In vs 40 Daniel says that a 4th kingdom would arise that would rule the world. Who was this? The Bible does not identify this power, however history tells us that the Greek empire was overthrown by the Roman in 168 BC. It was the Romans who ruled the world at the time of Jesus.