I am a vegetarian, but don't know much about the history of this diet. Do you know anything about it?
Me too! I know that its roots are from India and ancient Greece.
That would explain why there's so much delicious vegetarian food in those countries! 
Oh totally! Us vegetarian basically practice the non-consumption of any animal meat including sea animals. But some vegetarian will eat eggs and dairy products.
I do like my cheese, but none of the other stuff. I think I'm a lacto-vegetarian. Is that correct?
Yes that is right. :) The ancient greeks and Indians practiced vegetarianism for religious purposes with the idea of non violence toward animals. 
I totally get that - I love animals. The ancient Greeks and Indians sound as if they were good people.
Yes, but it is nice that the tradition is coming back. In the 4th - 6th centuries following the Christianization of the Roman Empire vegetarianism nearly disappeared from Europe.