Is an acid (just) a proton donor and if not, what is the correct definition of an acid?
Your teacher said that this is not completely correct as there are (at least) three common definitions for acids and bases (Wiki has a good article for this https://en.wikipedia.org/wiki/Acids_and_Bases):


Arrhenius: In water, acids under this definition protonate the water and form Hydronium (H3O+). Bases in this definition deprotonate the water.
Brønsted–Lowry: This is your proton donor definition. Acids in this definition donate protons to bases.
Lewis: In this definition acids are electron pair acceptors and bases are electron pair donors. 


The relationship between these definitions is that as chemistry evolved we ran into more chemicals (acids and bases) with similar reactivities that the previous definitions didn't quite encompass and so chemists broadened the definitions. So, as you go down the list above you will find more and more chemicals that match your new definition of acid or base.