Actually, rejey's definition is stricter than usual.
Basically, a vegetarian doesn't eat or drink any part of any dead animal. That's it. Milk and eggs aren't part of the animal, so consuming them, either alone or as ingredients in something else, is considered acceptable for vegetarians. Some vegetarians don't know about things like gelatin or rennet, or aren't concerned about them, but technically, those things are not vegetarian.
I'm not sure why rejey included clothing items like leather in the definition, since the word vegetarian is strictly a word to describe a diet, and nothing else.
As for vegan, that's pretty easy to define, too. Vegan is a non-dietary term for someone who doesn't eat/drink/use anything that comes from any member of the animal kingdom in any way. This includes milk, eggs, honey, fur, leather, silk, etc, or any products including even trace amounts of any of these things.
But the most important thing to remember here is that the label isn't important. If you want to be a lacto-ovo veg, which means you still consume eggs and milk and things made from them, then that's ok. If you want to eliminate milk, but keep eggs in your diet, that's your decision, and that would be called an ovo-vegetarian. Many people here are leaning towards vegan, but not quite there yet.
The most important thing is making the transition go smoothly so you don't get frustrated and give up just because you broke one of the "rules".