Grothendieck developed in EGA a number of notions of smoothness for a scheme and, more generally, for a morphism of schemes. For algebraic varieties over a field, one already had a classical notion of a nonsingular variety.
A scheme of finite type over a field $k$ is smooth if after extension of scalars from $k$ to the algebraic closure $\bar{k}$ it becomes a regular scheme, i.e. the stalks of its structure sheaf are regular local rings in the sense of commutative algebra.
A relative version of a smooth scheme is the notion of smooth morphism of schemes.
Specifically, a finitely presented commutative associative algebra $A$ over a field $k$ is smooth if either of the following equivalent conditions holds
the $A$-module $\Omega^1_k(A)$ of Kähler differential forms is a projective object in $A Mod$;
with $A$ regarded as an $A$-bimodule in the evident way, it has a projective resolution.
For commutative $k$-algebras a discussion is for instance around theorem 9.1.2 in
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