What does it mean to reduce a $p$-adic number modulo $p^n$ to an element of $\mathbb{Z}/p^n\mathbb{Z}?$ I think it means to take the $n^{th}$ coordinate of our $p$-adic number. Would I be right?
This is the context. Let $Z_p$ denote the $p$-adic numbers. If $f \in Z_p[X_1, \ldots, X_m]$ then we can obtain a corresponding polynomial in $\mathbb{Z}/p^n\mathbb{Z}[X_1, \ldots, X_m]$ by reduction modulo $p^n.$
Yes, to reduce a $p$-adic integer modulo $p^n$ you take the final $n$ digits of its $p$-adic expansion.
For example, as a 5-adic number, $\frac{1}{3} = \dots313132$.
So ...
$\frac{1}{3} \mod 5 = 2_5 = 2 \quad$ and $ \quad 2 \times 3 = 6 = 1 \mod 5$
$\frac{1}{3} \mod 25 = 32_5 = 17 \quad$ and $ \quad 17 \times 3 = 51 = 1 \mod 25$
$\frac{1}{3} \mod 125 = 132_5 = 42 \quad$ and $ \quad 42 \times 3 = 126 = 1 \mod 125$
$\frac{1}{3} \mod 625 = 3132_5 = 417 \quad$ and $ \quad 417 \times 3 = 1251 = 1 \mod 625$
etc.