What is a counting number? The definition of natural number is given as The counting numbers {1, 2, 3, ...}, are called natural numbers. They include all the counting numbers i.e. from 1 to infinity. at the link http://en.wikipedia.org/wiki/List_of_types_of_numbers. 
What does counting number mean, are there any numbers which are not countable?

The counting numbers are $\mathbb{Z}^{+} =  \big\{1, 2,3....,∞\big\} $ also known as the natural numbers.
Even though the counting numbers go on on forever, they can be counted. 
Given a set $X$,
$X$ is denumerable or enumerate if there is a bijection $\mathbb{Z}^{+} → X$. 
A set is countable if either it is finite or it is denumerable. 
Thus the set $\mathbb{Z}^{+} =  \big\{1, 2,3....,∞\big\} $, is countable since there is a bijection $\mathbb{Z}^{+} → \mathbb{Z}^{+}$. 
This gives the listing $\mathbb{Z}^{+} =  \big\{1, 2,3....,∞\big\} $
The set of real  $\mathbb{R}$ numbers  is uncountable (not countable) since there is no bijection  such that  $\mathbb{R}^{} → \mathbb{Z}^{+}$. 
For example, $∞$ cannot be map to 1, since infinity is not a number. 
Thus there has to be a one to one and surjective relationship between the set of the natural numbers with the given set of numbers.
