What was Daniel's interpretation of Nebuchadnezzar's first dream?
Nebuchadnezzar saw a giant image that looked very powerful and shone brightly. Seeing this image frightened and bothered him, but a king of his magnitude and stature does not frighten easily (Daniel 2:1). The head of the image was made of fine gold; the chest and its arms were made of silver. It's belly and thighs were made of brass, its legs were made of iron and its feet were made of iron and clay.
"You, O king, are a king of kings. For the God of heaven has given you a kingdom, power, strength, and glory; and wherever the children of men dwell, or the beasts of the field and the birds of the heaven, He has given them into your hand, and has made you ruler over them all—you are this head of gold. But after you shall arise another kingdom inferior to yours; then another, a third kingdom of bronze, which shall rule over all the earth. And the fourth kingdom shall be as strong as iron, inasmuch as iron breaks in pieces and shatters everything; and like iron that crushes, that kingdom will break in pieces and crush all the others. Whereas you saw the feet and toes, partly of potter's clay and partly of iron, the kingdom shall be divided; yet the strength of the iron shall be in it, just as you saw the iron mixed with ceramic clay. And as the toes of the feet were partly of iron and partly of clay, so the kingdom shall be partly strong and partly fragile. As you saw iron mixed with ceramic clay, they will mingle with the seed of men; but they will not adhere to one another, just as iron does not mix with clay."
King Nebuchadnezzar was chosen by God to become the first ruler. He thought of himself as the king of kings of the present world. His mighty kingdom was symbolized by gold and it is the head of the image, which indicates importance. It was not the most powerful kingdom but it was the richest and its government was a total monarchy. Nebuchadnezzar was given authority over the entire earth, not only the territory of his own immediate empire but he did not administer authority to rule over the nations and the world's population, as he did not have the means for such an undertaking. But when the future one-world ruler, the Antichrist, comes to power during the Tribulation he will have the means to rule the entire world and he will.
The second kingdom of silver (Medo-Persian) and the Greek Empire of brass are briefly mentioned. The rulers of those kingdoms were given authority to rule over the entire world but they did not. Medo-Persian Empire was a limited monarchy, the ruling kings were bound by the laws they enacted (Daniel 6:8, 12, 15). The third kingdom, the Greek Empire, did begin an absolute monarchy with Alexander the Great, but upon his death his kingdom was divided into four parts, which in time developed into a democracy, the weakest form of government that changes with the whim of the people and the elitists who control the majority of the people.
The fourth kingdom of iron and clay was the most powerful of all the kingdoms. The Roman Empire was an imperial form of government and was prophesied to break into pieces and subdue all things, yet it did not fulfill this prophesy during its reign from 27B.C. to 1453 A.D. It will break into pieces and subdue all the nations of the earth after the one-world government is formed in the last days (Daniel 7:23-24). Then the Antichrist will rise to power (Daniel 7:24):
'The fourth beast shall be
A fourth kingdom on earth,
Which shall be different from all other kingdoms,
And shall devour the whole earth,
Trample it and break it in pieces.
The ten horns are ten kings
Who shall arise from this kingdom.
And another shall rise after them;
He shall be different from the first ones,
And shall subdue three kings.
The dream specified that the fourth kingdom would be a divided kingdom. This took place in 395 A.D. after the death of the Emperor Theodosius 1 (379-395). The western division headquartered in Rome fell in 476 when Romulus Agustus was forced to abdicate to the Germanic warlord Odoacer.  The eastern division (Byzantine Empire) headquartered in Constantinople, Turkey ended in 1453 with the death of Constantine XI and the capture of Constantinople by Mehmed II, the leader of the Ottoman Turks. 
Mixing of iron and clay stood for the fact that the Roman Empire would be strong, but also weak. This was a strong indication that it was the most powerful empire but it collapsed from within due to moral and political corruption.
The last part of the interpretation, "As you saw iron mixed with ceramic clay, they will mingle with the seed of men; but they will not adhere to one another, just as iron does not mix with clay" (2:43) is thought to be by some to represent real rulers of the Roman Empire are fallen angels. In Daniel chapter ten we can see that the true rulers of the Persian and Greek empires were fallen angels:
"Then he said, "Do you know why I have come to you? And now I must return to fight with the prince of Persia; and when I have gone forth, indeed the prince of Greece will come" (Daniel 10:20).
The angel sent to Daniel to give him the answer to a question he had asked God was delayed for 21 days by the Prince of Persia (Daniel 10:2, 13); that prince was not Cyrus or any of his officers, it was a fallen angel. No man has the physical strength to delay an angel, consider how one angel killed 185,000 soldiers in one night (2 Kings 19:35). If the term "they" are fallen angels it means that sometime in the reign of this fourth empire the fallen angels could possibly mate with women as they did in the days of Noah (Genesis 6:4) and after the flood (Numbers 13:33; Deuteronomy 3:11, 1 Samuel 17:4, 2 Samuel 21:16-22). Some theologians speculate that the Antichrist will be the offspring of a fallen angel, possibly Satan himself.
Endnotes  Isaac Asimov, Asimov's Chronology of the World, Harper Collins, 1989, p.10  Asimov, p.198.