When did practice of being a Vegan start? Did Vegans exist in the 1800s, or is it a 20th/21st century invention?

Donald Watson coined the term vegan in 1944 when he co-founded the Vegan Society in England. At first he used it to mean "non-dairy vegetarian", and by May 1945 vegans explicitly abstained from "eggs, honey; and animals' milk, butter and cheese".

Interesting. I think veganism may have existed before too, in Asian cultures (I don't mean chinese, I mean Indian subcontinent).

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There have always been strict vegetarians, who kept to a diet that would now be described as vegan, but the word vegan was not coined until the 1940s.

Political veganism, ie protesting and angrily screaming about cakes having cow bone gelatin and generally having a continuous hissy fit over the rights of animals started with straight-edge hardcore music in the 1990s. Finn Mickinty on Youtube covers that scene and it's influences well.

Not eating or being involved with animal products has been a thing for a long time. Hinduism comes to mind where even leather shoes are frowned upon. But also, I interact with daily with a few dear friends who just don't like meat. I just don't like sweets. I'm sure many other's have both preferences and avoidance for their own reasons.

My over-educated theory on why is that the agriculture and hunter-gatherer societies did not cleanly merge. Case in point, Micronesians live off Taro root, Alaskans live off meat, and storable and travel-worthy wheat is what made Britain so powerful over the centuries. (Read Guns, Germs, and Steal).

"When did practice of being a Vegan start? Did Vegans exist in the 1800s, or is it a 20th/21st century invention?"

The practice of veganism actually got a full start in the mid 1940's or as stated in one answer, in 1944, when the term vegan was coined. However the roots are from vegetarianism, which dates back over twenty five hundred years ago. Veganism was an idea isn't of the Jain which dates back somewhat later. However Jainism allowed for the wearing  of wool, as it didn't cause harm. However many vegans disagree with that position. However modern veganism didn't get its real leg up, if you will, until 1957, when vitamin B12 became commercially available as a supplement. Without that  and/or foods that are fortified with it, few vegans or even strict vegetarians would last more than about five years. Some make it longer, and a rare few are actually able to live out a relatively long life without taking in more. But that's where the roll of genetics comes into play.

The word vegetarian was coined in 1842. The concept of being vegetarian has been around even longer. For instance, early Greeks had Pythagorasians. We are not sure but maybe the early Pythagorasians ate fish. So maybe technically not vegetarian. 

 there were vegetarians for religious reasons in early India and China. 

One could argue that these early "vegetarians" were also the precursors to modern-day vegans. Their non-violent stance on animals seems more vegan-ish to me.  

If you don't get B12, you die.

B12 only naturally come in animal products.

Being "vegan" only happened in circa 1950 when Gene-Splicing enabled people to vat grow micro-organisms to produce B12.  Finding facts is difficult in the Internet, especially since the search engines no longer allow a Boolean Search to NOT. 

It started in the 1800s. You can look it up on Wikipedia.

It was invented in our time