Vegetarians do not consume fish or fish-products.
A person who eats fish (and other seafood), but does not consume mammals and birds, is a Pescetarian, not a Vegetarian.
However, it is not uncommon for people to become Pescetarian prior to making the transition to Vegetarianism.
However, Pescetarianism is not considered a valid Vegetarian diet.
If you'd like to know more about Vegetarianism and why Vegetarians don't eat fish, here is a great place to start.