Depends on the vegetarian.

"Ovo-lacto-vegetarians" eat eggs and cheese.

Quite a few vegetarians of the South Asian diaspora will refuse to consume eggs too, on the premise that it is a "potential life." Milk, cheese, and honey may be okay, because life doesn't have to be taken to harvest those. I've had friends that would refuse birthday cake cause it was baked with eggs.

Vegans, as you said, consume no animal products, so no milk, no cheese, no honey.