What does it mean "A oxidizes on B"
This answer is a summary of the string of very good comments to the OP's original set of short questions.  This is both my take on the questions and an attempt to produce a more readable "net" answer than those contained within the set of individual comments.

Regardless of the intended context, "A is oxidized on B" means that A looses electrons.  It's seems this context likely implies that B is an electrode surface "on" which A is being oxidized.  

Regardless of context, if A is oxidized by B, then B is reduced.  

If B is an electrode surface, then the electrons from A needn't reduce B itself, but could flow through that part of the cell (the anode).

I hope this (correctly!) answers each of the OP's questions.