Below I am going to rewrite a definition which I found here.
$\textbf{Definition:}$ Let $A, B\subseteq \mathbb{R}^n$. Then, $A$ and $B$ are said to be $\textbf{congruent}$ (Euclidean metric) iff there exists an isometry (a distance preserving function) $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$ with $f(A)=B$.
$\textbf{Question:}$ What does $f(A)=B$ mean here? $\underline{Specifically}$, how do we define $f(A)$? I know $f(A)$ has a very specific meaning in Topology which I can't remember.
$f(A)$ is the image of the set $A$, which is the set of values which $f$ takes on elements of $A$. In symbols, $f(A)=\{f(a):a\in A\}$.
This definition is in no way specific to topology - it originates from set theory and is used all over the place in mathematics. Sometimes, to disambiguate between image of an element and an image of a subset, notation $f[A]$ is used instead.