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?

Counting numbers are numbers you can use to count whole items. You can use the positive integers to count marbles, for example. Whether 0 is a counting number or not, that's a big debate you may or may not be interested in. As another answerer so amusingly noted, fractional and complex numbers are unsuitable for counting live animals.
As to whether numbers of a particular kind are themselves countable, that's a different issue that requires different words: you need to ask about countable numbers, not counting numbers. For example, there are infinitely many rational numbers between 0 and 1 and you can count them in a theoretical sense even though you can't count them in a practical sense. I would start with $\frac{1}{2}$, continue with $\frac{1}{3}$, $\frac{2}{3}$, then $\frac{1}{4}$, $\frac{3}{4}$, $\frac{1}{5}$, $\frac{2}{5}$, etc. In short, I can give you a concrete algorithm that could theoretically count all these numbers.
There are also infinitely many irrational numbers between 0 and 1. I would start with $-1 + \sqrt{2}$. But what about $\pi - 3$? What about all those transcendental numbers we don't have symbols for? Even if I was a genius (which I am not), I couldn't give you an algorithm to count those numbers.
