Short Answer: The toes ("ten" is not specified) are treated as synonymous with the feet, which are mentioned in relation to the legs of iron. ("Toes" are not even mentioned in the recounting of the dream itself.) The significance is that they were part iron and part clay, which signified division within the kingdom, part of it being as strong as iron, and part as brittle as common clay. This division would characterize the fourth kingdom.
Since this is a hermeneutics site, let's walk through the text of Daniel 2:40-43 and do some exegesis. Regarding the "iron legs" we first read the following:
Then there will be a fourth kingdom as strong as iron; inasmuch as iron crushes and shatters all things, so, like iron that breaks in pieces, it will crush and break all these in pieces.
This is the first sentence in the paragraph that describes the "iron legs." We learn here that (A) we are looking at a fourth kingdom, (B) it will destroy the previous three kingdoms, and (C) that the iron signifies destructive power. The paragraph continues...
In that you saw the feet and toes, partly of potter’s clay and partly of iron, it will be a divided kingdom; but it will have in it the toughness of iron, inasmuch as you saw the iron mixed with common clay. As the toes of the feet were partly of iron and partly of pottery, so some of the kingdom will be strong and part of it will be brittle.
There are several very important things to note here:
this is a description of the same kingdom; there is no indication in the text that this is a fifth kingdom or a latter state of the fourth kingdom. To say otherwise is to make more of the vision than the author, Daniel, and God did!
this is a divided kingdom because of the mixture of materials -- not because it has "toes"! There is no indication here that the "toes" should be itemized as ten (twelve?) separate segments or anything like that. Every mention of division is accompanied by the explanation that it has two materials.
the mixture of iron and clay indicates the kingdom will be divided in that part of the kingdom will have the toughness of iron, and the other part will not (in fact, it will have the opposite; the brittleness of common clay pottery)
The paragraph continues:
And in that you saw the iron mixed with common clay, they will combine with one another in the seed of men; but they will not adhere to one another, even as iron does not combine with pottery.
This comments on the actual combining of the two parts of the kingdom. You cannot really combine iron and clay. Likewise, the strong, iron-like parts of the kingdom and the brittle, common clay-like parts of the kingdom will "combine with one another in the seed of men", but they will not actually adhere to one another. The two parts are "in bed together," so to speak, but are not actually united.
The fourth kingdom of Nebuchadnezzar's dream in Daniel 2 is as strong as iron, and will crush all the other kingdoms. Thus, it will rule the world. (cf. 37-39) However, it will be a divided kingdom, because not all of it will be as strong as iron; part of it will be as brittle as common clay pottery. The two parts of the kingdom will be in bed together, but they will not adhere to one another.
Rome? Perhaps. I'm thinking of the strength of their military and the weakness of their senate? I'll save the specific speculations for another day, or another interpreter, but hopefully this exegesis will be a helpful first-step in the right direction.