Vegetarianism is the practice, or the belief system, of not eating meat or fish. It is the name for the whole idea, not for a person — a person is a vegetarian. Because it ends in -ism, it works like a topic word: you talk *about* vegetarianism, you argue for or against it, you write an essay on it. It is a fairly formal, written-sounding word; in everyday speech people simply say "I'm vegetarian" or "she doesn't eat meat". People choose it for different reasons — health, animals, religion, the environment — and the word covers all of them without saying which. In practice most vegetarians still eat eggs and dairy; not eating any animal product at all is veganism, a stricter thing with its own word. The word is uncountable, so no 'a' and no plural.