Your question, as it seems to me, is mainly a nomenclature issue.
Class, in mathematics, means usually something described by a formula: all things with a certain property.
Equivalence classes, for example, are called classes but could just as well be sets. In fact sets are also classes in the context of set theory. You can see this in other places as well (e.g. conjugacy class).
In set theory the Russell paradox (and various other paradoxes as well) showed that not every collection is a set. This is why the notion of a class was invented. It turns out that this notion is (at least philosophically) close to other notions of class in mathematics.
Formally, however, note that there are "normal functions; normal spaces; normal distributions; etc." but many of these are somewhat orthogonal and have nothing to do with one another. Classes of set theory are constructs in set theory, while conjugacy classes are constructs in group theory (used elsewhere as well, of course).
So to sum up, the word "class" is commonly used to describe a definable collection (definable from what? well, that depends on the particular context); and as Qiaochu remarked, in a very technical sense every set is a class anyway (in the set theoretical context of the word).