Answer by Michael Chan:
"a" refers to anything of that type, nothing specific.
"the" refers to a specific something of that type, nothing else. It is usually used when referring to something that both parties already know of.
I am a vice-president in the company.
In this case, I am probably one of the two or more vice-presidents in that specific company that we have been talking about previously.
I am the vice-president of the bank.
In this case, there is no other vice-presidents, there is only one, which makes me a lot more "big-shot" in that particular bank we have been talking about, than any banks which have tons of vice-presidents all over the place.