A node on a curve is a singular point that locally looks like the intersection of two lines. I think the precise way to say this is that $p \in X$ is a (closed?) point on a scheme $X$ (of finite type over a field $k$?), then the completion of the local ring at $p$, $\widehat{\mathcal{O}}_{X,p}$ should be isomorphic to $k[[x,y]]/(xy)$ (the completion for the intersection of two lines).

The first question is whether this is correct.

The second question is whether you get the n-dimensional version of a node by requiring the completion of the local ring to look like intersection of $n+1$ coordinate planes: $k[[x_0,x_1,...,x_n]]/(x_0 x_1\cdots x_n)$.

The third question is: what exactly does it mean for such a singularity to be isolated? This is easy to imagine over $\mathbb{C}$ (there is analytic neighborhood containing no other singularity), but how to say this in something like the Zariski topology?