In Daniel 2:31-36 Daniel recounts Nebuchadnezzar's dream of the statue to him. After the statue is described, Daniel says the following:
You continued looking until a stone was cut out without hands, and it struck the statue on its feet of iron and clay and crushed them. Then the iron, the clay, the bronze, the silver and the gold were crushed all at the same time and became like chaff from the summer threshing floors; and the wind carried them away so that not a trace of them was found. But the stone that struck the statue became a great mountain and filled the whole earth.
Later, we read the interpretation, which God explained to Daniel, Daniel relayed to Nebuchadnezzar, and [someone] recorded in Scripture for us. Here is the interpretation of the stone, from Daniel 2:44-45:
In the days of those kings the God of heaven will set up a kingdom which will never be destroyed, and that kingdom will not be left for another people; it will crush and put an end to all these kingdoms, but it will itself endure forever. Inasmuch as you saw that a stone was cut out of the mountain without hands and that it crushed the iron, the bronze, the clay, the silver and the gold, the great God has made known to the king what will take place in the future; so the dream is true and its interpretation is trustworthy.”
It is worth remembering that the purpose of the latter passage is to interpret the significance of the dream. The dream is not open to personal interpretation; it meant something, and the latter passage explains what the meaning was.
Several things are immediately clear from the dream and its interpretation:
The entire statue was crushed all at once. This is not a picture of a gradual conquest.
When the kingdoms of the statue were crushed, "not a trace of them was found"; this is not a picture of a gradual infiltration, or a parallel reign of some kind, and there is no allowance for any of the statue kingdoms to continue in their existence. They were crushed, became like chaff, and were carried away by the wind.
It is the kingdom set up by God that crushes and puts an end to the other kingdoms. So, to answer your question, the "stone cut out without hands" is the kingdom set up by God.*
The kingdom set up by God "filled the whole earth," "will never be destroyed" or replaced, and will "endure forever." The picture is clear: God's kingdom will rule the entire earth -- forever -- in place of the prior kingdoms of the statue.
* It is worth noting that the king and the kingdom are indistinguishable in this dream (or most anywhere in Scripture, for that matter); cf. 38, 44.
The stone cut without hands is the kingdom of God, which would suddenly and completely obliterate all of the kingdoms of the statue and rule in their place over the entire earth, forever.