A natural number, which can also be called a counting number, is represented by the digits from 1, 2, 3 through to infinity. The number 0 is included if natural numbers are defined as non-negative integers, but not if they are defined as only positive integers. In math, there must be an infinite number of natural number digits, since each natural number is defined in part by having a number that follows it. These numbers are also whole numbers, not fractions or decimals, and can be used for counting or ordering.
The main distinction between a natural number and an integer is that natural numbers, with the exception of zero, are only positive. There is no number below zero, and a natural number can't be followed by zero, such as is the case with -1,0. Essentially this defines natural numbers as anything zero or above that is whole and not fractional. Zero is generally considered to be the only natural number that is not positive.
The concept of zero evolved long after civilizations began using counting numbers. Earliest records of counting numbers from 1-10 date to over 4000 years ago, when the use of specific written code to signify place were used by the Babylonians. The Egyptians wrote hieroglyphs for each digit, but it wasn't until about 1000 BC that the concept of zero was created by the Mayan and Olmec civilizations.
Though the Olmec and Mayan groups show the first records of the use of zero, the concept of zero also developed in India, in the 7th century BCE. It was the Indian use, rather than Mesoamerican use that was adopted by civilizations like the Greeks.
There are many ways in which natural numbers can be used in math applications. They can limit problems by suggesting that the answer must be a natural number. They are also studied in specific application in set theory, mathematics that evaluates sets of things. Number theory may evaluate natural numbers as part of the set of integers or independently to see if they behave in certain ways or exhibit certain properties.
Perhaps one of the widest uses of natural numbers comes to us very "naturally." When we are young we learn to count from 0 onward. Even young children can easily begin to learn the difference between one and two, or explain how old they are. This study continues as children begin school and learn to manipulate natural numbers, how to multiply, divide, add and subtract them. Only after the concept of natural numbers is learned is the concept of integers introduced, and the possibility of negative numbers, which can confound some kids at first, is usually learned in fourth or fifth grade at earliest.