There are some vegetarians which do not eat any animal flash at all, including fish like salmon, and then there are others who eat fish and poultry but stay away from beef and other similar types of meat. There are certainly different types of vegetarians out there and so there are no universal rules as to what a vegetarian cannot eat, though this type of diet usually does exclude animal meat of most kinds.
It is up to the individual what to eat and sometimes that includes fish along with poultry. There is also veganism which excludes all types of animal meat and even animal byproducts such as milk or cheese. Those who follow this type of diet usually do not eat any type of fish or animal flesh at all.