Maybe make a table with overview. Possible sources: FGA book.
Many notions, such as smooth, flat, etale, proper, open immersion, can be defined in the much more general contexts of Toen. See for example HAG II, the Barcelona notes, and the Brave new algebraic geometry paper.
http://mathoverflow.net/questions/15474/functorial-characterization-of-morphisms-of-schemes
From review of Murre lectures on fundamental gp: A flat morphism is open. A faithfully flat morphism is an effective epimorphism.
For reference, we collect here some basic definitions of various kinds of morphisms of schemes.
“E.g. any morphism between smooth k-schemes”
nLab page on Morphism of schemes