What do you call people that eat fish but not meat?
Vegetarians don't eat any animals including fish. Those who don't eat meat but fish are called pescetarians. For practical reasons, sometimes pescetarians say that they are vegetarians, as Joe says in his answer in this question:


  I read something recently (might've been a blog, might've been some
  online publication), explaining why the person called themselves a
  vegetarian and yet would eat fish:
  
  
  Because it was easier than explaining all of their can/can't eat items.
  
  
  Many people aren't familiar with the term 'pescatarian', and if
  someone invites you to a dinner party, it's easier to just say 'I'm a
  vegetarian' than 'I'm a lacto-ovo pescetarian' and have to take the
  time explaining what it means. If they prepare vegetarian food, you
  can eat it ... they don't need to know the whole ontology of different
  types of vegetarianism.