No-one truly knows but there are many stories. Japan, England and even Ireland lay claim to the game but it is so old and it has evolved over so many years, nobody can really say who invented it.
The Origin Of Chess

India is one of those countries who had a chess-like game in the pre-chess days which contributed to what chess is today. That's probably what your teacher was referring to. Since the pawn was not yet allowed to move two spaces on the opening move, the practice of developing your bishop to b2, g2, b7, g7 is still referred to as an "Indian Bishop." I'm pretty sure I also heard that their rooks were represented by elephants rather than castles.
I too, similar to Billium248, have heard it all started in India. And not to beat the dead horse, but I have always had a lot of respect for all the "Indian" openings such as KID or KIA, for it's been played for so many years. I actually almost beat a a chess engine on master - the hardest engine it could offer - with QID.
I know, it's all amazing isn't it? Before I'd heard it may have originated in India I'd heard - and assumed - somewhere in Europe; somewhere like England, Spain, or Northern Europe.

Actually, like theniceman, I've heard that the only thing we know for certain is that no one knows for certain where the first chess-like game originated. My guess is that as the world shrank with global travel, all of the Indian, Japanese, Chinese, English, Irish, etc. variations got assimilated and refined into what we now call chess.
There's always something else interesting about chess. There's a group on here specfically dedicated to the history and development of chess. I'm sure they can give you plenty of details.
And here's something else I find interesting: Did you know that there is currently a Planet Earth vs Outer Space chess game going on?
http://main.uschess.org/content/blogcategory/198/475/
I was at one of the Chess Team meetings at school, and our coach just happened to metion that chess is Indian -- not Native American, the other kind of Indian -- is that true? If not, what is it?