I'm not sure whether a single class refers to a single subset of a larger set, or whether it refers to multiple subsets of a larger set (i.e. a set of sets)
Which is it? If it's the latter (a single class refers to multiple subsets of a set), how do you refer to a single subset within the class?
Mathematicians use several terms when they are referring to zero or more (potentially many, or even infinitely many) of an object. Some such terms are collection, class, and set.
For more information about the difference between sets, classes, and collections, see Asaf Karagila's answer on the matter. Here is a brief summary:
Mathematicians use the term "collection" informally. It refers to zero or more, potentially many, of some object. There are no other restrictions. For example, you could have a collection of matrices, or a collection of integers, or even a collection of sets. As Asaf says:
Think of it as a big bin full of trash, diamonds and empty bottles of beer, it doesn't have to make sense what is in this collection, it's just a collection.
Using the term "collection" means we aren't being formally precise. "Classes" and "sets" are both kinds of collections, where we are more careful to define what we mean.
A set is precisely defined; there are strict rules about what is a set. For example, $\varnothing$ (the empty set) is a set; $\{\varnothing\}$ (the set containing one element which is the empty set) is a set; and the set of integers and the set of real numbers are also sets.
Not every collection of objects is a set; in fact, there are specific rules about what is allowed to be a set (for example, the power set of a set is a set), and nothing else is allowed to be a set.
We can form a set of some sets (or a set of sets of sets), using the rules. But sometimes we want to consider ALL sets, and not just some of them. This requires a more general term, a class.
A class is zero or more sets that share a defined property. For example:
The class of all sets with an even number of elements
The class of all sets with no elements
The class of all subsets of $\{1, 2, 3\}$
The class of all sets which do not contain themselves (all sets $x$ such that $x$ is not an element of $x$)
As long as you can write down a description of which sets are included, and which sets are not -- then you have defined a class.
Any set is a class: for example, if $x$, $y$, and $z$ are sets, then "the collection of three elements, $x$, $y$, and $z$" is both a set and a class. It's a set (written $\{x,y,z\}$) and it's also a class because it is given by some description of which sets are in the class. In this case, exactly $3$ sets belong to the class.
Not all classes are sets, however -- just because you can write down a description of which sets are included, does NOT make your description a set. For example, the collection of all sets is a class, but it is not a set.
The fact that not all classes are sets is a result of a paradox called Russell's paradox, which says that there is no such thing as "the set of all sets that don't contain themselves." It is easy to write down a description of "all sets that don't contain themselves" -- so the collection of all such sets is a class -- but Russell's paradox says that it's not a set.
According to Nagel and Newman, authors of ``Godel's Proof"---
pg. 16 ``Let us understand by the word `class' a collection or aggregate of distinguishable elements, each of which is called a member of the class."
Later, on pg. 23, they expound a little bit further by declaring, ``and the occurrence of these contradictions (antinomies that have emerged with Cantor's theory of infinite numbers) has made plain that the apparent clarity of even such an elementary notion as that of $class$ (or $aggregate$) does not guarantee the consistency of any particular system built on it.''
Then after adding: "Since the mathematical theory of classes, which deals with the properties and relations of aggregates or collections of elements, is often adopted as the foundation of other branches of mathematics, and in particular for elementary arithmetic, it is pertinent to ask whether contradictions similar to those encountered in the theory of infinite classes infect the foundations of other parts of mathematics." The authors point out that Bertrand Russell constructed a contradiction within the framework of elementary logic analogous to the one first developed in Cantor's theory of infinite classes, and then proceed to introduce ``Russell's antinomy" by stating that
$Classes$ $seem$ $to$ $be$ $of$ $two$ $kinds:$ $those$ $which$ $do$ $not$ $contain$ $themselves$ $as$ $members,$ $and$ $those$ $which$ $do.$
And then they go on to elaborate further.
