TITLE: Sets and classes
QUESTION [17 upvotes]: I don't quite understand the difference between sets and classes.
A class consists of sets. Why is not a class a set?
A group itself is a set. A isomorphic class of group is not a set, right?
Could you explain how to distinguish a set from a class.

REPLY [12 votes]: In addition to all that has been written here, and before (see links below), let me give you the most informal intuition on this subject.
Consider the natural numbers $\Bbb N$. Each proper initial segment is finite, and in fact represents a particular finite number. But there is no number represented by $\Bbb N$. Why? Because $\Bbb N$ is too large to be a natural number. Natural numbers are finite, but collections of natural numbers don't have to be finite.
In modern set theories such as ZFC, the objects of the universe are sets. And every set has a size, or cardinality. Classes are those collections of sets that do not have a size. Much like $\Bbb N$ is an initial segment which is too big to be finite, classes are collections which are too big to be sets. (One remark is that in some set theories classes are objects and have measurable sizes, but those go beyond the scope of ZFC and naive set theory.)
Some links, where many words have been written:

difference between class, set , family and collection
Difference between a class and a set
What should we call the 'sets' which don't exist under certain set theory axioms?
Does $A=\{a|\forall x\in \emptyset\ H(x,a) \}$ make sense?