Counting is the mathematical action of adding (or subtracting) one, usually to find out how many objectss there are or to set aside a desired number of objects (starting with one for the first object and proceeding with a one-to-one correspondence); however, counting is also used (primarily by children and linguistic students) to demonstrate knowledge of the number system. Counting sometimes involves numbers other than one (for example: when counting money or when "counting by fives" -- 5,10,15,20,25...); but, relatively simple numbers (most always integers) are still used.
There is archelogical evidence that humans have been counting for, at least, 50,000 years [1 - p.9]; the development of counting led to the development of mathematical notation and numeral systems. Counting was primarily used, by ancient cultures, to keep track of economic data; such as debts and capital.
See also: enumerative combinatorics
References
1 -- An Introduction to the History of Mathematics (6th Edition) by Howard Eves (1990)