Cardinality

The fundamental concept in the theory of infinite sets is the cardinality of a set. Two sets A and B have the same cardinality if there exists a mapping from the set A onto the set B which is one-to-one, that is, it assigns each element of A exactly one element of B. It is clear that when two sets are finite, then they have the same cardinality if and only if they have the same number of elements. One can extend the concept of the “number of elements” to arbitrary, even infinite, sets. It is not apparent at first that there might be infinite sets of different cardinalities, but once this becomes clear, it follows quickly that the structure so described is rich indeed.

http://plato.stanford.edu/entries/set-theory/