Lacto-Ovo Vegetarian: Do not consume meat or fish but do consume milk and eggs. This term can also be broken down to Lacto Vegetarian or Ovo Vegetarians which denotes eating only or the other.
Strict/Pure Vegetarian: Someone who does not consume meat, eggs, or dairy but may use products tested on animals or products made of animals. This is a diet not a lifestyle. Most people who fit into this category are vegetarian for health not for animals or the environment.
Vegan: People who do not consume any meat, fish, dairy, or eggs. They also do not consume processed foods with ingredients derived from animals such as cassien or gelatin. Vegans also avoid purchasing and using products that contain animal products or have been tested on animals. Veganism is often more a way of life than simply a diet.
Raw Vegan: People who follow a vegan diet plus not consuming food that has been cooked at temperatures above 116 F or 47 C.
And a couple that I don't know too much about
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The only diets that I have personal experience in are vegetarianism and veganism.
Veg*n is term used to denote both vegetarian and vegan.