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 is smooth if after extension of scalars from to the algebraic closure 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.
For commutative -algebras a discussion is for instance around theorem 9.1.2 in