Let $X$ be the affine variety in $\mathbb C^3$ (coordinates $a, b, c$) defined as the zero set of $ac - b^2$. The variety has a "double point singularity" at the origin.
This is somewhat intuitive, but I don't know how to make sense of singularities for more complicated examples.
For example, let $Y$ be the affine variety in $\mathbb C^4$ (coordinates $a, b, c, d$), defined by $ac - b^2$ and $bd - c^2$.
What kind of singularities does this have?