Digit
A digit is any one of the ten symbols (0-9) used to write numbers in the base-10 number system.
Definition
A digit is one of the ten symbols we use to write numbers: $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, and $9$; every number is made from these ten digits. In the base-$10$ positional numeral system more generally, each digit's value depends on its position (place value) within the numeral, and the number of digits in a positive integer $n$ is $\lfloor \log_{10}(n) \rfloor + 1$. Extending to any base $b$, a digit is any element of $\{0, 1, \ldots, b-1\}$, and the value of a numeral $d_n d_{n-1} \ldots d_1 d_0$ in base $b$ is the sum of $d_k \cdot b^k$ for $k$ from $0$ to $n$.
Example
The number $472$ has three digits: $4$, $7$, and $2$; the number $1{,}000$ has four digits. In $3{,}058$: the digit $3$ is in the thousands place (value $3{,}000$), $0$ is in the hundreds place, $5$ is in the tens place (value $50$), and $8$ is in the ones place (value $8$). Divisibility rules derive directly from these place values: a number is divisible by $9$ if and only if its digit sum is divisible by $9$, so for $729$: $7+2+9=18$, and $18/9=2$, confirming $729 / 9 = 81$.
Key Insight
Just like the $26$ letters of the alphabet can make every word in English, the $10$ digits can make every number in our number system. The word "digit" comes from the Latin "digitus" (finger), because humans historically counted on fingers, and different bases use different digit sets: binary uses only $\{0,1\}$, hexadecimal uses $\{0\text{-}9, A\text{-}F\}$. The choice of base $10$ is anatomical, not mathematical: base $12$ (dozenal) would make divisibility by $3$ and $4$ as easy as divisibility by $2$ and $5$ is in base $10$.