Terminology for a subset that matches an element As a strict beginner, I have been warned about getting mired in the natural language terms "inclusion" and "belonging," and this makes perfect sense to me. However, I have a problem about the characterization of "elements" and "subsets" which is related but harder for me to resolve.
For the following set:
X = {{1, 2}, 1, 2}
We have the power set:
P(X) = {{{1, 2}, 1, 2}, {{1, 2}, 1}, {{1, 2}, 2}, {1, 2}, {{1,2}}, {1}, {2}, {}}
So {1, 2} belongs to X and is also included in X. However, it seems to me that the {1, 2} which belongs to X is somehow different than the {1, 2} that is included in X. Note that the {1, 2} that belongs to X is an element of that set, while the {1, 2} that is included in X is a subset of that set. Is it worthwhile to note this difference, or is there even terminology for this difference? In my title I have said that they "match" but this seems to me to be an inadequate characterization.
 A: This situation comes up very often in set theory (considered as a subject in its own right), since there the only objects we have are sets. That is, whenever we have a set $x$, all of its elements are sets as well.
So, we might have sets $x$ and $y$ such that $x\in y$ and $x\subseteq y$. But $x$ is still the same set, whether considered as an element of $y$ or as a subset of $y$. So, in your example, $\{1,2\}$ is $\{1,2\}$ either way. Both sets have the same elements (1 and 2), so are the same set.
In answer to your question then: They are the same set. The interesting property of being both an element and a subset doesn't have a particular name, since it's relatively simple to just write $\{1,2\}\in X$ and $\{1,2\}\subseteq X$.
But there is some terminology for something closely related. Sometimes the situation occurs that you have a set $x$ with the property that every member of it is also a subset of it. In this case, we say that $x$ is transitive. 
A: You're completely correct. You ask "is it worthwhile to note this difference?" and my answer would be "yes, that's a good thing to have noted for yourself." But in practice, most sets we encounter in math tend to contain just one kind of thing (either numbers, or sets of numbers, or functions, or ...), so this kind of potential confusion seldom arises. It's great that you noticed it, though -- shows you're thinking hard about each new definition you encounter. 
A: Following J.P.'s considerations, it can be useful to think to an example of not-pure set theory (I say pure set theory when the only objects of the "universe" are sets).
Take an army : it is a set of brigades; every brigade in turn is a set of companies; every company in turn is a set of platoons; every platoon in turn is a set of soldiers.
But soldiers a nor more "sets of" : they are urelements.
So, each soldier is member of a platoon, of a company, and so on. But soldiers are not subset of anything (in this "model").
