Thanks to many large animal organizations, coupled particularly with poorly informed celebrity spokes-people, and widely perpetuated misinformation by non vegans like Oprah, there is a very wide misunderstanding of what veganism is. Veganism is seen by many as a diet -- someone is vegan if they don't eat animal products. Some describe it in even looser terms then that, defining a vegan as someone who doesn't eat much in the way of animal products, or doesn't eat animal products most of the time, such as Bill Clinton. It is because of this association of vegan as a diet that we often see "vegetarian/vegan" or "vegetarian and vegan", two words that have very little in common lumped together, which further adds to the confusion.
So let me try to clarify a little bit for people. Since diet is where people often confuse the issue, let us start there: veganism is not a diet.
Someone who is vegan eats a vegetarian diet, but unlike more loose interpretations of "vegetarian", a vegan does not eat any animals or animal products such as eggs, dairy, D3, sugar produced with bone char, alcohol clarified with isinglass, fish, etc. In other words, vegans stick with the original definition of vegetarian which was "no animal products".
So, if veganism is not a diet, the next question is... what is it? That answer is so simple that it is often hard for people to fully comprehend it without giving it lot of thought. Veganism is a commitment to nonviolence, starting with what we wear, what we use to entertain ourselves and what we eat.
The question then is if "vegetarian" can also mean someone who does not eat any animal products, then why did Donald Watson feel the need to coin a new term?
The answer to that can be found in the very first newsletter published by the Vegan Society: "'Vegan News' (Quarterly Magazine of the Non-Dairy Vegetarians) No.1":
We should all consider carefully what our Group, and our magazine, and ourselves, shall be called. 'Non-dairy' has become established as a generally understood colloquialism, but like 'non-lacto' it is too negative. Moreover it does not imply that we are opposed to the use of eggs as food. We need a name that suggests what we do eat, and if possible one that conveys the idea that even with animal foods taboo, Nature still offers us a bewildering assortment from which to choose. 'Vegetarian' and 'Fruitarian' are already associated with societies that allow the 'fruits'(!) of cows and fowls, therefor it seems we must make a new and appropriate word.
So, as we can see, in 1944, for many the word "vegetarian" had already come to mean a diet that included the use of animal products.
Shortly after, the Vegan Society fleshed out the definition of vegan as such:
Veganism is a way of living which excludes all forms of exploitation of, and cruelty to, the animal kingdom, and includes a reverence for life. It applies to the practice of living on the products of the plant kingdom to the exclusion of flesh, fish, fowl, eggs, honey, animal milk and its derivatives, and encourages the use of alternatives for all commodities derived wholly or in part from animals.
That means that we seek, as much as possible, to avoid exploiting, using and killing others, nonhuman as well as humans.
It is simple, but it flies directly in the face of everything that our societies do.
If you pay attention, almost every person that you see walking down the street is exploiting nonhuman animals in some form or shape, from leather, wool, silk and fur clothing and footwear, to eating their flesh or secretions. Further, this mindless disregard for the well being of others most often extends to human animals as well, through more obvious things like sweatshop labour that went in to the clothes that we wear to child slave labour that went into the cheap chocolate that we eat, and to less obvious things like the violence that we display to all the underprivileged groups in our society through acts of racism, sexism, heterosexism, classism, ableism, etc.
Veganism, as a practice of ahimsa, has the potential to address, and help redress, all of these issues.
Some see this as a "hardship" or something that is very difficult to do in our current society, but is it not much harder to know that you are needlessly contributing to the pain and suffering of others? Is it not harder to live with yourself knowing that you are causing harm that is completely avoidable? But even more importantly, is it not harder to be locked up, be forcibly impregnated only to have your offspring taken away from you year after year so that someone can steal your milk? Is it not harder to be brutally killed to satisfy someone's greedy palate? In other words, it may sometimes be an inconvenience for us, privileged species that we are, but it is always harder for the person who is actually being exploited, abused, murdered.
And, it let us be clear, it is not because it is necessary, but because we enjoy it. Because as a species, we take pleasure in those products, regardless of how they got to our supermarket shelves and onto our plates, or onto our bodies.
We have choices, and we have the power to stop contributing to the exploitation of others with every action, purchase and decision that we make. It is time that we started exercising that power.
It starts with you.
^  At the moment, I can only find secondary sources for this quote. If anyone knows the primary source, and even better has a link to it, I would greatly appreciate it! A similar, but different enough definition is included in the UK Vegan Society's Articles of Association and reads as such:
"veganism" denotes a philosophy and way of living which seeks to exclude — as far as is possible and practicable— all forms of exploitation of, and cruelty to, animals for food, clothing or any other purpose; and by extension, promotes the development and use of animal-free alternatives for the benefit of humans, animals and the environment.