I define a vegetarian as one who avoids eating animal flesh, bones, and other body parts. Fish are animals so I do not consider fish to be a vegetarian food. Gelatin is made from animal bones so that is also not a vegetarian food. Vegetarians may however eat eggs and dairy products.
I define a vegan as one who avoids eating all animal products, including dairy and eggs. Many who call themselves vegan, including me, make an exception for honey, which while produced by an animal does not originate from an animal's body, technically. Many vegans also avoid using products derived from animals, such as fur, leather, wool, and silk. Though I do prefer to buy clothing that is not derived from animals, I am not strict about this, and do not consider it part of the essential definition of being vegan.
As far as reasons, the primary reasons many people state for avoiding animal products boil down to health, ethics, the environment, and religion. My own reasons contain some of each, other than religion (I am an atheist, though I have Buddhist leanings).
Health: I recognize that people are omnivores and can survive and thrive on all kinds of diets, but I believe a diet based primarily on plant products can be beneficial. From my research I consider a strictly vegan diet to contain all nutrients necessary for good health other than vitamin B12, which I take as a weekly supplement. (B12 is actually made by microorganisms in the soil, not animals, but modern hygienic practices strip much of this out of our food, so for strict vegans it is necessary to supplement.)
Being vegan or vegetarian is definitely not a guarantee of being healthy or maintaining a healthy weight. Coke and potato chips are vegan, after all, and here in San Francisco all kinds of highly-processed fake meats and dairy products are readily available. I base my diet on high-starch foods like whole grains, beans, and tubers, per the McDougall Program recommendations. McDougall recommends reserving animal products for "feast days" such as birthdays and holidays. I currently prefer to stay vegan even then, but enjoy more processed foods with higher sugar and fat content. I've had some really amazing vegan desserts!
Ethics: I am not an animal rights activist; I do not believe non-human animals have an inherent right to life. However, I personally could not bring myself to kill an animal unless I were truly starving, so I don't feel right eating them. If someone wants to go shoot an elk and have elkburgers for his family to eat all winter, I have no objection to that; it actually might be friendlier to the environment and might kill a smaller number of animals (see next section) than eating a purely plant-based diet. But I have no interest in hunting or fishing, myself.
As far as dairy and eggs, cows and chickens are treated horribly on American factory farms. I suppose I wouldn't object to truly free-range eggs and milk gathered from family farms, but I've never liked eating eggs, and it would be a lot of trouble to make sure that every dairy-containing product I wanted to eat contained milk from a "happy" cow. So I prefer to just avoid these products completely, at least to the best of my ability. (I'm sure I've unintentionally eaten milk and eggs in breads and other foods served at restaurants and parties from time to time. I'm not going to beat myself up over that.)
Some who are otherwise vegan eat animals such as shrimp and clams with the assumption that their mental facilities are not developed enough to feel any pain. I think this is a slippery slope, as some say that chickens and cows are basically dumb and bred for food. Especially since I've never liked seafood anyway, I find it easier to just avoid eating anything from the animal kingdom, and at the same time recognize that it is impossible to exist without causing any harm or taking some life along the way, however unintentional.
Environment: American factory farming has done great damage to the topsoil and water supply. Some of this applies to plant crops as well as animal farming, true. And even farming plants takes animal life, in the form of field mice and other native creatures destroyed and/or displaced. But on balance, from my research I feel that choosing to eat a diet lower in animal products is better for the environment. As I've never liked meat or eggs and already stated the problems with dairy, I prefer to just avoid eating foods from the animal kingdom altogether. True locavores have my respect, whether they eat meat or not, but I for one will not give up my bananas!
These are my personal definitions and reasons for being vegan, which have certainly changed over time, and will doubtless continue to evolve. Food choices are a controversial topic, but worth thinking about.