Here is an answer for the affine case. Assume $f:A\to B$ is a homomorphism of Noetherian rings. Then $B$ is a filtered colimit of smooth (finitely generated) $A$-algebras iff $f$ is regular (flat with geometrically regular fibers). This is due to Popescu and Spivakovsky; see for instance Teissier's Bourbaki talk
http://www.math.jussieu.fr/~teissier/documents/Approx.BBk.pdf
and the references therein (the above result is Thm. 1.1).
If $A=k$ is a field, this says that $B$ is a colimit of smooth $k$-algebras iff it is geometrically regular over $k$. If $k$ is perfect (e.g. the prime field!) you can remove "geometrically". The field case may be simpler than the general case: this is unclear to me.

Here is an answer for the affine case. Assume $f:A\to B$ is a homomorphism of Noetherian rings. Then $B$ is a filtered colimit of smooth (finitely generated) $A$-algebras iff $f$ is regular (flat with geometrically regular fibers). This is due to Popescu and Spivakovsky; see for instance Teissier's Bourbaki talk
http://www.math.jussieu.fr/~teissier/documents/Approx.BBk.pdf
and the references therein (the above result is Thm. 1.1).
If $A=k$ is a field, this says that $B$ is a colimit of smooth $k$-algebras iff it is geometrically regular over $k$. If $k$ is perfect (e.g. the prime field!) you can remove "geometrically". The field case might possibly be simpler than the general case.