[summary: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, …
The natural numbers are the numbers we use to count things.]
A natural number is a number like 0, 1, 2, 3, 4, 5, 6, … which can be used to represent the count of an object. The set of natural numbers is $~$\mathbb N.$~$ Not all sources include 0 in $~$\mathbb N.$~$
Natural numbers are perhaps the simplest type of number. They don't include negative numbers, fractional numbers, irrational numbers, imaginary numbers, or any of those complexities.
Thanks to their simplicity, natural numbers are often the first mathematical concept taught to children. Natural numbers are equipped with a notion of [-addition] ($~$2 + 3 = 5$~$ and so on) and [-multiplication] ($~$2 \cdot 3 = 6$~$ and so on), these are among the first mathematical operations taught to children.
Despite their simplicity, the natural numbers are a ubiquitous and useful mathematical object. They're quite useful for counting things. They represent all the possible cardinalities of finite sets. They're also a useful [data_structure data structure], in that numbers can be used to [numeric_encoding encode] all sorts of data. Almost all of modern mathematics can be built out of natural numbers.
[todo: Add a "formalization" lens with the Peano axioms. I recommend at least one page with just the raw Peano axioms (and very little prose), and another gentler introduction sort of like http://bit.ly/29glDrR and http://bit.ly/29nKYAL, albeit more to-the-point and probably without going all the way up to non-standard number territory.]