Let f : $X \rightarrow Y$ be a morphism of smooth schemes of finite type over $k$. Then I want to understand the meaning of a morphism is "smooth of relative dimension $d$".
In particular if Z is a smooth, closed subscheme of a smooth scheme X of codimension $d$ then is it true that inclusion map from $Z$ to $X$ is smooth of relative dimension $d$.
Smooth is the algebraic geometry counterpart of "submersion". Over the complex numbers, it corresponds to fiber bundles, so that the fiber has dimension $d$. In general, it is still true that for a smooth surjective morphism the dimension of the fiber is constant, this is $d$.
Also, smooth implies flat and a closed immersion is never flat unless it's also an open immersion.