Each set can be viewed as a class, but the converse is not true. You can have sets of sets, but not necessarily sets of classes.
A class is basically any collection you can think of; this was the approach used by Cantor in his initial theory (the naive set theory). However, it soon became apparent that attempting to build a set theory on that basis led to contradictions; one of these contradictions is Russel's paradox.
To solve that problem, modern set theory uses an axiomatic method: there is a set of axioms (the standard system is called ZFC) that define a limited set of operations that can be used to construct sets. The collection of objects that satisfy a property is a class, but, unless you can construct it using the axioms of the theory, you cannot assume that it is a set; in particular, you cannot use it as a member of a set.
A class that is not a set is called a proper class. For example, the class of all sets is a proper class: there is no set of all sets.