Let $f:X\to Y$ be a flat, surjective morphism of $k$-schemes with connected fibres i.e. $f$ is a family.
Definition: Let $T$ be a $k$-scheme. A deformation of $f$ (over $T$) is a family $g:\mathfrak D\to Y\times _{\text{Spec} k} T$ with the following property: there exists $t_0\in T$ such that the family $\mathfrak D_{t_0}\to Y\times \{t_0\}$ is isomorphic to $f:X\to Y$ (i.e. the``square diagram'' has two isomorphisms).
I don't understand what are formally the schemes: $\mathfrak D_{t_0}$ and $Y\times \{t_0\}$. Would you help me?
For instance, if $f:X\to Y$ is a morphism of $k$-schemes I know that the fibre of $y\in Y$ is formally defined as $$X_y:=X\times_{\text{Spec }k}\text{Spec}\,k(y)$$ I'd like a similar description for the aforementioned objects.
The family $g$ is assumed to be a morphism over $T$. The structure map $\mathfrak D\to T$ should be part of the data of the family, and $Y\times_kT$ is a $T$-scheme via second projection. Now given $$g:\mathfrak D\to Y\times_{\textrm{Spec }k}T,$$ every closed point $\textrm{Spec }k(t_0)\to T$ induces a $k(t_0)$-morphism $$ g_{t_0}:\mathfrak D_{t_0}:=\mathfrak D\times_T\textrm{Spec }k(t_0)\to (Y\times_{\textrm{Spec }k}T)\times_T\textrm{Spec }k(t_0)=Y\times_{\textrm{Spec }k} \textrm{Spec }k(t_0), $$ by base change. The latter is just pedantic notation for your $Y\times_{\textrm{Spec }k}\{t_0\}$. Here is your diagram, I suppose:
$$\require{AMScd} \begin{CD} \mathfrak D_{t_0} @>{\simeq}>> X\\ @V{g_{t_0}}VV @VV{f}V \\ Y\times_k \textrm{Spec }k(t_0) @>{\simeq}>> Y. \end{CD}$$