Natural Number
A natural number is a positive counting number (1, 2, 3, ...) used to count and order objects.
Definition
Natural numbers are the numbers you use when counting: $1$, $2$, $3$, $4$, $5$, and so on, starting at $1$ and continuing forever. Formally, $\mathbb{N} = \{1, 2, 3, \ldots\}$ denotes the positive integers, closed under addition and multiplication; many textbooks instead use $\mathbb{N}$ to mean $\{0, 1, 2, \ldots\}$, writing $\mathbb{N}^+$ or $\mathbb{Z}^+$ when $0$ is excluded. The natural numbers can also be constructed from first principles using the Peano axioms: there is a first element ($1$ or $0$), and every element has a unique successor, with addition and multiplication defined recursively from the successor function.
Example
Counting students in a class, $1, 2, 3, \ldots, 28$, every count is a natural number, and the sum or product of any two naturals is always a natural number, such as $7 + 12 = 19$ or $4 \times 9 = 36$, though subtraction and division can leave the set ($3 - 5 = -2$, $7 / 2 = 3.5$). By the Fundamental Theorem of Arithmetic, every natural number greater than $1$ is either prime or a unique product of primes. Proof by strong induction lives entirely in $\mathbb{N}$: to prove $P(n)$ for all $n$, show $P(1)$ holds, then show that if $P(k)$ holds for all $k < n$, it holds for $n$.
Key Insight
They are called "natural" because counting is the most natural thing humans do with numbers, coming before all the other kinds of numbers we invent. Kronecker's famous quote captures this hierarchy: "God made the integers; all else is the work of man." The naturals are the foundation on which all other number systems ($\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$) are constructed, and in category theory the resulting structure is the initial object among semirings.