When most people think of vegetarians, they think of lacto-ovo-vegetarians: People who do not eat beef, pork, poultry, fish, shellfish, insects or animal flesh of any kind, but do eat eggs and dairy products are lacto-ovo vegetarians (“lacto” comes from the Latin for milk, and “ovo” for egg). This is the most common type of vegetarian in North America.
In other words, a vegetarian is someone who doesn't eat any kind of meat. A vegetarian doesn't eat chicken, hamburgers, steak, fish, shrimp, lobster or any animals or seafood.
The word itself can be used as either a noun, as in "That person is a vegetarian," or an adjective used to describe the diet, or way of eating, as in, "That person follows a vegetarian diet."
Lacto-vegetarian is sometimes used to describe a vegetarian who does not eat eggs, but does eat dairy products. Many Hindu vegetarians are lacto-vegetarians who avoid eggs for religious reasons while continuing to eat dairy.
Ovo-vegetarian refers to people who do not eat meat or dairy products but do eat eggs. Some people are ovo-vegetarians because they are lactose-intolerant.
There is some debate, of course, as to what, exactly constitutes a vegetarian diet, and what makes one a vegetarian. For example, many people insist that eggs are not vegetarian, and many people, particularly in the UK, exclude certain types of cheese from a vegetarian diet as well.
If you'd like to learn more about this, here's a few resources which can help:
So What Exactly, Do Vegetarians Eat?
You'll notice that a vegetarian diet is most often defined by what it excludes, that is, what vegetarians don't eat, rather than what they do eat, which may leave you wondering, so, what do vegetarians eat?
A vegetarian diet includes just about everything (other than meat, of course!) you can imagine, from slurpees to wheatgrass juice: beans, legumes such as lentils, whole grains such as rice, wheat and quinoa, baked goods and breads including cookies, cakes, croissants and bagels, all vegetables, squash such as pumpkin and butternut squash, fruits, nuts and sea vegetables such as seaweed, processed and chemical foods such as Twinkies, MSG and high-fructose corn syrup, coffee, tea, alcohol, eggs and eggs products (usually), dairy and dairy products (usually) such as ice cream, cheese, milk, cream, butter, cream cheese, soy food products such as edamame and tofu, meat substitutes such as veggie burgers and seitan, all oils, herbs and spices, most seasonings such as soy sauce, Nama Shoyu and hot sauce, and, well, you get the idea!