In algebraic geometry, a morphism f : X → S {\displaystyle f:X\to S} between schemes is said to be smooth if (i) it is locally of finite presentation (ii) it is flat, and (iii) for every geometric point s ¯ → S {\displaystyle {\overline {s}}\to S} the fiber X s ¯ = X × S s ¯ {\displaystyle X_{\overline {s}}=X\times _{S}{\overline {s}}} is regular. (iii) means that each geometric fiber of f is a nonsingular variety (if it is separated).
In algebraic geometry, a morphism f : X → S {\displaystyle f:X\to S} between schemes is said to be smooth if (i) it is locally of finite presentation (ii) it is flat, and (iii) for every geometric point s ¯ → S {\displaystyle {\overline {s}}\to S} the fiber X s ¯ = X × S s ¯ {\displaystyle X_{\overline {s}}=X\times _{S}{\overline {s}}} is regular. (iii) means that each geometric fiber of f is a nonsingular variety (if it is separated).
Source: Wikipedia "Smooth morphism" · CC BY-SA 4.0
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