= Solution
For a point $y\in Y$ with <residue field> $\kappa(y)$, the <scheme-theoretic fibre> is the <base change of a morphism of schemes>
$$
\boxed{X_y=X\times_Y\operatorname{Spec}\kappa(y).}
$$
The map $\operatorname{Spec}\kappa(y)\to Y$ is the canonical point map. This <fibre product of schemes> retains the <structure sheaf>, including any nilpotents, rather than just the set $f^{-1}(y)$. For an <affine scheme> map induced by $A\to B$, at $\mathfrak p\in\operatorname{Spec}A$ it is
$$
X_{\mathfrak p}=\operatorname{Spec}(B\otimes_A\kappa(\mathfrak p)).
$$
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