The standard example of a deformation is a proper flat map $f:X \longrightarrow S$ between schemes. The terms infinitesimal or local characterize the base $S$.
A local base like $S = Spec \ k[t_1,...,t_n]$ often serves as an affine neighbourhood of a base point. $S$ has many closed points. So you can compare fibres $X_s = f^{-1}(s), s \in S$, over different base points $s \in S$ as well as the variation of the fibre $X_s$ when $s$ moves through $S$.
The other extrem is a base like $S = Spec \ \mathbb C$. Here you study a single fibre of $f$ on its own.
The intermediate steps are infinitesimal neighbourhoods of a single fibre $X_s$: You choose
$S = Spec \ k[t_1, ..., t_n]/^k, k \in \mathbb N,$
an infinitesimal neighbourhood of the point $s$. The most simple non-trivial case is $S= Spec \ k[t]/$, the double point. The family of all infinitesimal neighbourhoods is the formal neighbourhood of $s$. Extending information about $f$ from the formal neighbourhood of a base point to a local neighbourhood starts with comparison theorems for base change.
One of the first examples of building bottom-up a deformation via infinitesimal neighbourhoods is the construction of a versal deformation of a compact complex manifold $X_0$ with $H^2(X_0, \Theta_{X_0}) = 0$ by Kodaira-Nirenberg-Spencer.