What does it mean "A oxidizes on B" I have just managed to understand the difference between oxidation and reduction. So basically if A oxidises B, then B loses electrons, right? But what does this formulation mean: "A is oxidized on B"? What is this "on" suppose to mean? That A is losing electrons to B or vice-versa? 

Sorry if this sounds like a very stupid question, but I'm really confused by presence of "on something"

EDIT:
If you search for this


  "Carbonate-based electrolytes commonly used in Li-ion batteries are
  reduced on negative electrodes"


on Google, you will find the first paper and this specific formulation.

And here you can find the paper: http://pubs.acs.org/doi/abs/10.1021/acs.jpclett.5b01727?journalCode=jpclcd
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.