A vegan buys things that have no ties what-so-ever to animals or fish. No eggs, milk, cheese, yogurt, fish, etc.
A vegetarian will not eat meat or fish.
A lacto-vegetarian will not use dairy or meat but may eat eggs.
An ovo-lacto vegetarian will not eat eggs, dairy, or meat.
Hope this helps, I am a vegetarian and do not eat meat, but will eat fish...