In short, vegetarians do not eat anything made from dead animals. For some reason, some people seem to think vegetarians eat chicken or fish. A chicken is an animal, not a vegetable. A fish is also not a vegetable. Therefore, vegetarians don't eat either of those things.
Some vegetarians are vegans.
Vegans, in addition to not eating animals, don't eat products obtained from animals, including dairy products and eggs. Many vegans also don't eat honey. Some foods that look vegetarian have hidden animal products, such as vegetable soups made with beef broth, or things fried in lard. Vegetarians also don't eat these things. There's more I could say about this, but it's been better said elsewhere.