The Pigeonhole Principle states that if you have $n$ pigeons and $n-1$ pigeonholes, then at least one of those holes must contain at least $\lceil{\frac{n}{n-1}}\rceil$ many pigeons. So if you have $3$ pigeonholes and $11$ pigeons, then there is at least one hole with at least $3$ pigeons. That's a more informal definition.
A proper mathematical definition of the Pigeonhole Principle in my textbook states:
If $f : X \to Y$ is a mapping and $|X| >|Y|$, then there is a $y \in Y$ with $|f^{-1}(y)| \geq 2$.
The last part of this definition is a bit tricky for me to understand. So what it's saying is that; if we have a mapping from $X$ to $Y$ and the number of elements in $X$ (the number of pigeons) is larger than the number of elements in $Y$ (the number of pigeonholes), then there exist an element (pigeonhole) in $Y$ which has $2$ different preimage elements.
Is my understanding of the definition correct?
PS - The definition and explanation are translated from German, therefore some of the words may sound a bit weird, such as "preimage". In German it's "Urbild", which refers to the inverse image of an element in a set.