The easiest example I can think of is to consider the map $\mathbb{A}^1\to \mathbb{A}^1$ by sending $x\to x^2$, where $x$ is the coordinate. Then take the fibre over the origin, which would be $Spec k[x]/(x^2)$. The idea is that $x=0$ in the fibre should be counted "twice".
The easiest example I can think of is to consider the map $\mathbb{A}^1\to \mathbb{A}^1$ by sending $x\to x^2$, where $x$ is the coordinate. Then take the fibre over the origin, which would be $Spec k[x]/(x^2)$. The idea is that $x=0$ in the fibre should be counted "twice".