Short answers, pulled from the story.
A natural number is one of the numbers 0, 1, 2, 3, and so on, used for counting and for labeling the result of a count. They are also called positive integers, non-negative integers, whole numbers, or counting numbers, and the set is denoted in bold or blackboard bold.
Whether 0 is a natural number is a matter of convention. Most early authors excluded it, but mathematicians including Bertrand Russell, Nicolas Bourbaki, Paul Halmos, and John Horton Conway included it, and the inclusion of 0 gained wider adoption in the 1960s and was formalized in ISO 31-11 in 1978.
A cardinal number describes the size of a finite collection, as in there are seven days in a week, while an ordinal number labels a position in an ordered series, as in the third day of the month. Both uses draw on natural numbers but answer different questions: how many versus which position.
The five Peano axioms state that 0 is a natural number, every natural number has a successor, 0 is not the successor of any natural number, numbers with equal successors are equal, and the axiom of induction carries a truth from 0 through every successor. Named for Giuseppe Peano, they describe natural numbers without explicitly defining what they are.
The natural numbers are a semiring, also called a rig, because they are not closed under subtraction and lack additive inverses. Subtracting a larger natural number from a smaller one gives a negative number outside the set, so the structure fails the requirement to be a ring.
Nicolas Chuquet used the phrase progression naturelle, meaning natural progression, in 1484, and the earliest known use of natural number as a complete English phrase appears in 1763. The 1771 Encyclopaedia Britannica defined natural numbers in its article on the logarithm.
In the standard set-theoretic construction due to John von Neumann, 0 is the empty set and the successor of any set is formed from that set, with the natural numbers being the intersection of all inductive sets. In this construction every natural number n is a set containing n elements, each a natural number less than n.