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In informal usage, a cardinal number is a number used in counting (a Counting Number), such as 1, 2, 3, ....
Formally, a cardinal number is a type of number defined in such a way that any method of counting Sets using
it gives the same result. (This is not true for the Ordinal Numbers.) In fact, the cardinal
numbers are obtained by collecting all Ordinal Numbers which are obtainable by counting a given
set. A set has 
(Aleph-0) members if it can be put into a One-to-One correspondence with the finite
Ordinal Numbers.
Two sets are said to have the same cardinal number if all the elements in the sets can be paired off One-to-One. An Inaccessible Cardinal cannot be expressed in terms of a smaller number of smaller cardinals.
See also Aleph, Aleph-0, Aleph-1, Cantor-Dedekind Axiom, Cantor Diagonal Slash, Continuum, Continuum Hypothesis, Equipollent, Inaccessible Cardinals Axiom, Infinity, Ordinal Number, Power Set, Surreal Number, Uncountable Set
References
Cantor, G. Über unendliche, lineare Punktmannigfaltigkeiten, Arbeiten zur Mengenlehre aus dem Jahren 1872-1884.
Leipzig, Germany: Teubner, 1884.
Conway, J. H. and Guy, R. K. ``Cardinal Numbers.'' In The Book of Numbers. New York: Springer-Verlag, pp. 277-282, 1996.
Courant, R. and Robbins, H. ``Cantor's `Cardinal Numbers.''' §2.4.3 in
What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.
Oxford, England: Oxford University Press, pp. 83-86, 1996.