A
set X is said to be
finite if there exists a
bijection f:X->{1,2,3,...,n} for some
natural number n. Then, n is called the
cardinal of X, often denoted |X|.
Basically, a set is finite if you can count the number of elements in it. Above, we see a formal notion of counting in the bijection between the given set X and the set of the first n natural numbers.