terminology: set of sets What is the proper name for "a set of sets"? Is it just a "higher-order set" in general or a "secondary set" in particular? A Wikipedia link would be great. I've been unable to find a special term for "set of sets" there.
 A: As others have pointed out, technically, there is no distinction between a "set" and a "set of sets"; in fact, in most modern set-theoretic formalizations of mathematics, every object is a set, and so there is nothing a set can contain besides other sets.  A set such as $\{\{\{\emptyset\}, \emptyset\}, \emptyset \}$ contains two sets, one of which contains two sets, one of which contains a set.  In examples much more complicated than this, the notion of "second-order set" does not make much sense.  And in case you don't believe such sets come up, see Arturo's comment below.
However, for psychological reasons, other terms such as "family" and "collection" are often used.  (Royden's book Real Analysis describes a hierarchy for which terms he uses when; see the first paragraph of Chapter 1, Section 1 (p. 6 in my edition).)  This is similar to the psychological reasons that we choose to use different symbols (e.g., $\cdot$ or $+$) for group operations, depending on what sort of group we're looking at.
One last technical point: Logicians do sometimes work with "languages" in which every object is, say, a real number rather than a set.  In cases such as these, one needs second-order logic to talk about sets at all, and third-order logic to talk about sets of sets, etc.  However, most mathematicians work, more or less, in the "first-order language of set theory": every object is already a set, so there is no need for higher order sets.  (Although there are a few people who prefer second-order set theory as a formalization of mathematics, I think they are in a very small minority of the people who care about such things.)
A: In set theory, it is just a set.  We do not distinguish between the levels, as the elements of a set are sets, but they could be sets of sets of sets of ....  In fact, the usual ordinals have one set at each level (up to that ordinal)
A: In some contexts, a "set of sets" is called a family. Some examples:


*

*Extremal set theory: an intersecting family of sets.

*Matroids: a family of independent sets (but: a set of bases).

*Topology: a family of open sets.


This is semantic terminology. "Physically" (if you "believe" in ZFC) everything is considered a set, as mentioned by Ross and Qiaochu.
