Vegan - one who eats a diet consisting of no animals or animal products (meat, fish, eggs, milk products, honey), and, therefore, eats an entirely plant-based diet (fruits, vegetables, legumes, and grains). Often, vegans will also refrain from the use of any animal products (e.g., leather, fur) in their dress and lifestyle, as well.
Vegetarian - one who eats a diet consisting of no meat or fish. Vegetarians generally eat eggs, milk products, and honey, in addition to plant-based matter.
Lacto-ovo-vegetarian - a word often used to differentiate "vegan" from "vegetarian." A lacto-ovo-vegetarian eats milk products and eggs, but abstains from meat and fish. Some use the words "vegetarian" and "lacto-ovo-vegetarian" interchangably, although "lacto-ovo-vegetarian" is the more specific word.
Pescetarian - one who eats a diet that includes fish but NOT poultry, beef, or other kinds of meat. Pescetarians generally also eat eggs, milk products, and honey. Some use the words "vegetarian" and "pescetarian" interchangably, although "pescetarian" is the more specific word. Pescetarians who call themselves vegetarians are often criticized for not being "real" vegetarians because they eat animals (fish), although, of course, this is a matter of opinion.
Omnivore - one who eats a diet consisting of animals, animal products, and plants