A morphism of locally Noetherian schemes \(f \colon X \to Y\) is normal, respectively regular, if it is flat and for every \(y \in Y\) the fibre \(X_y\) is geometrically normal, respectively geometrically regular, over \(\kappa(y)\): \(X_y \times_{\kappa(y)} \Spec L\) is normal, respectively regular, for every finite field extension \(L/\kappa(y)\). A ring map \(A \to B\) of Noetherian rings is normal or regular when \(\Spec B \to \Spec A\) is.
A morphism locally of finite type is regular exactly when it is smooth.
If \(f \colon X \to Y\) is flat, \(Y\) is regular (respectively normal) and the fibres of \(f\) are regular (respectively normal), then \(X\) is regular (respectively normal).
For a Noetherian local ring \(A\), the completion map \(A \to \hat{A}\) is regular exactly when \(A\) has geometrically regular formal fibres; this holds for every local ring essentially of finite type over a field, and it is part of the definition of an excellent ring.
In characteristic \(p\), the Frobenius \(\Spec \FF_p(t^{1/p}) \to \Spec \FF_p(t)\) is flat and its only fibre is a field, but the fibre is not geometrically reduced: after base change to \(\FF_p(t^{1/p})\) it becomes \(\FF_p(t^{1/p})[x]/(x - t^{1/p})^p\). So this map is not regular, although both schemes are regular.
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