Being a vegetarian means eating plants and plant-based foods and dairy. Many vegetarians eat eggs but many others don't. In India, folks that don't eat eggs (but do eat dairy, go figure) call themselves "pure" vegetarians, and the folks that eat eggs are often called "eggiterians". But, in general, pure-vegetarians and eggiterians are considered to be vegetarians.

Thats pretty much it. Additions or subtractions from the above diet results in a different label (vegans, pescatarians, omnivores, buggyterians? etc.).