What are the infinitesimal normal neighbourhoods On page 72 of Mumford's The Red Book of Varieties and Schemes second, expanded edition (GTM 358), the last sentence of the first paragraph

Geometrically, the presence of nilpotents in $M_i$ should be taken as meaning that the $i$th point is surrounded by some infinitesimal normal neighbourhood.

As the author has mentioned, Spec$R$ consists of $n$ points with discrete topology. So I am confused by the "infinitesimal normal neighbourhood". Would you please tell me the meaning of this?
Thanks a lot.

The best way to understand Mumford's sentence is through an easy but non trivial example:     
Consider the closed subscheme of the affine line over a field $\mathbb A^1_k=\text {Spec}(k[t])$ given by the ideal $I=(t^{n+1}).$
This subscheme $V=V(I)\subset \mathbb A^1_k$ has as ring of global functions $R=k[t]/I$, a local artinian ring of dimension zero.
The subscheme $V=V(I)=\text {Spec}(R) $ (called the $n$-th infinitesimal neighbourhood of zero in $\mathbb A^1_k$) has only one point and nevertheless has a quite interesting  structure:    
Indeed, any global function on $\mathbb A^1_k$, namely a polynomial $P(t)=\sum a_it^i\in k[t]=\mathcal O(\mathbb A^1_k)$, can be restricted to $V$ and yields the function $\overline {P(t)}=\overline {\sum_0 ^n a_it^i}\in \mathcal O(V)=R=k[t]/(t^{n+1})$ on $V$.
The amazing power of scheme theory shines through: restricting a global function $P$ on the affine line to that single-point-scheme leaves enough information to reconstruct all the derivatives of  $P$ at the origin up to order $n$.    
This is a baby case demonstrating  one of the tremendous advantages of schemes over classical algebraic varieties.
The great Italian geometers of the beginning of the twentieth century had the intuition that they needed such infinitesimal neighbourhoods but didn't have the technical tools necessary to define and use them. And then came Grothendieck ...
