There's a wonderful quote which can be applied to situations such as this:

Never try to explain who you are to someone who is committed to misunderstanding you.

Occasionally you meet someone who is genuinely interested in your decision to become vegan and is open to change. Unfortunately those people are incredibly rare. More often than not, you will meet those who are invested in defending their own choices and in critising yours. Many people LOVE to argue their case given a chance. It's human nature. By entering into that you achieve nothing accept frustration and misery for yourself.

I use the tactic that others have mentioned. I don't bring up my choices in conversation unless it happens naturally or we are in a resturant for example. I make clear that I don't judge other people's choices and I simply focus on my own reasons. This approach can often inadvertently have far more influential affects than trying to argue your case with someone who has no interest in understanding and is keen to argue for the sake of it or to defend their own position.

You might want to pre-empt the whole "conversation" by avoiding the term "Vegan." "Vegan" seems to bring out hostility and almost anger in some people.

Instead you simply say you prefer to stick to meatless foods for your personal health.

No explanation, no defense, just "I feel healthier when I avoid meat dishes."

No one on earth can live an ethical and healthy life if they are putting into their stomachs the burnt, dead bodies of tortured and murdered sentient beings, or eating any foods that contribute to the suffering of those beings. If they are taking that into themselves they will become sick, both physically and spiritually. That usually gets the point across, and it's true. Cause and effect. This is important big stuff, never forget that.

I agree, drop the terms vegan and vegetarian. I use the term cruelty free diet, as that's what it is. Just don't expect anyone to change, and what they eat is their business. However, I support speaking out for those who cannot speak for themselves. If even one person in 10,000 becomes interested in a non dead body diet, my speaking out has been a success.

Get them to read 'Diet For A New America' (25th Anniversary Updated Version). Then they'll understand.
Either that or take them to a slaughterhouse!

To be honest, I don't think there's any one surefire method to convince people. Most find graphic images of slaughter offensive rather than educational. And I must confess that I didn't have any emotional reaction to watching slaughterhouse footage until well after I started eating a vegan diet. This isn't a matter of being a bad person, it's just that some people (like myself) are wired differently to not feel as many emotions or as strongly. (No water signs in this poster's chart!)

I started logically reevaluating my views and practices after watching [a VICE documentary on the Yulin dog festival which didn't take a sensationalistic approach like most vegan materials do. When people ask me why I do what I do, I send them this video first, because it really puts the Western view of animal slaughter in perspective.

EDIT: Tried to get the link right :-/

Well what exactly do you find unreasonable about the points they bring up? There are multiple sides to every story. Not to start a debate or anything, I just feel the need to give you some food for thought. Might not be vegan food though.

I am a meat eater. Specifically, my diet consists mostly of meat, dairy, nuts, fruit, tubers, and warm spices. Humans didn't evolve over 2 million years to be vegetarian or vegan.

We're animals and part of the food chain just like every other animal. You eat vegan because you feel compassion for animals, right? Is it wrong for a lion to eat meat too because the gazelle suffers? Or is it worse for the lion to suffer malnutrition on a grain-and-bean diet? I've never met a healthy vegetarian.

They say they feel good, but they look pale and weak. Your bodies react with sickness to foods outside your regular diet, not adapting to diverse or more readily available food. I prefer to eat organic and foraged but at least processed food doesn't make me sick.

The modern "paleo" diet is wrong too. Just like veganism, it restricts the diet too much. So much meat and not enough wild plants. Everyone in the food chaim, humans included, are in a symbiotic relationship with every other creature. An animal's dietary needs change with age, sex, moon phase, season....Everybody gets eaten by something.

That's the beauty of God, the beauty of birth and death, the cycles of nature. Every creature is both predator and prey. The lion eats the gazelle, the gazelle eats the grass, the grass "eat" the nutrients in the dirt, and the dirt is made by bacteria who eat all the waste. It's okay to be eaten.

Sure getting killed and eaten hurts an animal, but so does dying in every other way. Death is necessary for new birth. Energy and matter of every creature must be transformed, making room and providing the building blocks for new life. That is the beauty of a balanced self-sustaining ecosystem.

I'm an animal lover. One of those people who get really defiant if you say plants and other animals aren't people. But I'm morally okay with eating almost any creature. I do disagree with modern livestock farming, as well as any agriculture besides small homesteads. More of a hunter-gatherer type.

For most of human history, there were less of us and more other organisms. The food that grew wild was enough to sustain us. We would eat what we need, be eaten by something else, and thus give back to the Earth. Now there are too many of us and we only take without giving.

No modern lifestyle is free of cruelty to our fellow creatures. :(

I think veg has a rational ethical basis in a human society which is highly irrational, but no moral basis as a general rule. Of course if you take fish from the diet of an island people, or meat any nomadic group or jungle tribe, the 'right' becomes harmful.

This is the problem with righteousness: it's inherently unethical. Ethics have never in history come down to hard fast rules, but has always been a continual consideration of dilemmas, ad what's 'good' (beneficial) in one circumstance is often 'bad' (detrimental) in another.

The question is, hypothetically, if veg was made compulsory for everyone in the world, who would benefit (priviledged white people) and who would be harmed (remaining tribal communities). Veganism as an ideology is therefore an act of genocide, or at least, cultural domination.

I get what you mean, Gem
But who knows for sure if veganism/vegetarianism would harm the body-systems of tribal people? It may be detrimental to their culture and traditions....but would it harm their physical health?
Because of where they live and the options available to them then such people have no choice than to hunt and fish for a living. If they suddenly had options, and (theoretically) adopted a good balanced vegan diet, with all necessary nutrients, we can't really say without lifetime studies, how that would affect them physically.

I don't really see veganism/vegetarianism as exclusively a 'middle-class white-man's' eating fad, or any politically/culturally-motivated domination agenda....
Many Indian people are vegan/vegetarian.

I think the main point is that you dont have to explain yourself. Maybe even stop explaining it to yourself by putting yourself in the vegan box. Like who even cares. Why not just be an "I eat what makes me feel great" person. If you look healthy and feel great then I'll bet no one will argue against that.