(If you are vegetarian for purely environmental reasons, this is not about you. For those who don't know why one should become vegetarian for environmental reasons, there are a lot of studies that you can find at https://www.drawdown.org/ or https://www.greenamerica.org/climate-change-100-reasons-hope/top-10-solutions-reverse-climate-change etc or just read this Atlantic article).
There are three different levels of vegetarianism. The first one is very easy to apply: one doesn't want to use any body part of any animal; one doesn't want to eat body parts, does not want to wear leather, etc.
The second one is already a bit tricky: one does not want to kill any animal nor wants to pay for an animal to get killed (i.e. doesn't want to be directly responsible for the killing). It is hard to be consistent on this point. We routinely kill a lot of animals, from mosquitoes to termites. We kill useful spiders and snakes simply because they look ugly to us. We even kill rats, whose brain is so similar to ours that neuroscientists perform a lot of cognitive experiments on rats. We even kill fellow humans: criminals, terrorists, war enemies, ... And some people think that even abortion is a killing of human life. Killing is approved by society and by the vast majority of individuals in many situations. However, some societies have decided that the lives of some animals matter more than the lives of other animals. E.g., you cannot kill a cat the way you can kill a rat. In California you can't even kill a squirrel, which is a close relative of the rat, but you are encouraged to kill as many rats as you can. Vegetarians unilaterally decide that it is not ok to kill cows, chickens, fish, shrimps, snails and so on, but, presumably, are still ok with killing several other mammals (e.g. rats) and a lot of invertebrate animals (mosquitoes, termites, cockroaches...) We tend to empathyze with big animals like elephants and cows, but not with small animals like flies and worms; but there is no science that makes a big animal more important than a small one. So this view (that killing any animal is bad) is difficult to justify in a coherent way.
The third level is when one does not want to cause even indirectly the killing of an animal. This is the "vegan" level. The vegan argument goes like: cows are force-bred (serially "raped") annually to produce milk, their male offspring is slaughtered to become veal (about 200 million per year), their female offspring is raised to become other milk cows, etc. This level is also hardly consistent. First of all, the same argument applies as before: the vegan movement cares for the indirect cruelty and killing inflicted to some animals but not to others (e.g., when you destroy a spiderweb while dusting off a bookshelf you may kill the spider, when you wash your driveway you may flood an antnest). We eat vegetables that are grown in farms where farmers routinely exterminate all sorts of "pests", ranging from raccoons to birds. Secondly, in this case the killing is not necessary. In order to eat meat, someone must kill an animal. In order to eat an ice cream, it is not necessary that a cow's male calf gets killed. Unfortunately, it is the case that in most large and small farms, this is the practice; but "it is customary to do so" is not the same as "it is the only way to get it done". Eating an animal implies that the animal is killed. Using milk does not necessarily imply that anyone gets killed. The classic example to explain the difference is clothes: many clothes are made in sweatshops where children's life expectancy gets dramatically reduced, but that doesn't mean that wearing clothes is amoral, that we shouldn't wear any clothes at all. Just like it is very difficult to decide which clothes to wear, so it is very difficult to decide which cheese to eat. "Difficult" does not mean "impossible" though. If an accident kills 20 children in a sweatshop, you can accuse the people who bought clothes from that sweatshop of being morally responsible for those deaths, but you cannot accuse all the people who wear clothes of being responsible. Ditto for coal miners: if you live in Germany, chances are your energy comes from coal, and coal miners have a low life expectancy and often die young in accidents, but it doesn't mean that all users of energy in Germany are responsible for the deaths of all coal miners. The fact that a fundamental ingredient of smartphones is made in Congo, and funds the civil war there, is not a good reason to stop using smartphones. In all of these cases each of us has to ponder what we can contribute to improving the situation.
Furthermore, one has to be careful not to fall in the "naturalist fallacy", the idea that the natural world is better than the human-engineered world. It is debatable whether the cow would live a better life in its natural environment than the life it lives in an industrial farm. If we released all cows and bulls in their natural environment, they would be vulnerable to both predators and weather. Cows would be "raped" every day, and still get pregnant all the time. But in exchange they wouldn't get food, water nor a warm stable: they would get absolutely nothing. Calves would still move away within a few months, and many calves would be killed by predators (and sometimes even by jealous bulls). Nature is hardly idyllic. Hence, simply describing the horrible conditions of an industrial farm is not enough to make an argument against dairy products: you also have to prove that the horrible living conditions in an industrial farm are worse than the cruel destiny of a cow in the wild. (That argument works much better for horses: horses would indeed be able to live a happy life in the wild, and taming them for horse riding is a cruel sport that turns them into machines).
The frequent objection to vegetarians that plants are alive too is plain silly. Come back when you can prove that plants have feelings for their branches, their seeds, their fruits and their roots. We do know what animals feel for their bodies and for their children. Cutting branches helps trees grow stronger, but cutting limbs does not help animals grow stronger. Plants are not animals.
A more valid argument can be made against pets. Pets are actually treated very well by their owners, sometimes better than humans are treated in coal mines or factories. But one can argue that a pet has been kidnapped as a baby, separated from its parents and siblings, and reduced to a slave (in fact, to a piece of furniture or to a toy for children). You could do the same to a human baby and obtain the same results: if you kidnap a baby, the bay will grow up to love you and respect you, but i don't think this is moral (and, in fact, it is not legal anywhere in the world).
There are several clues that indicate that humans were not meant to be
Meat is the primordial fast food. At some point humans realized that they could absorb a lot of nutrients (notably, proteins) by just eating someone else's brain. Humans were cannibals before they were carnivores. The human brain was the first meat item in the menu (it is also one of the easiest to digest for non-carnivore). Animal meat must have looked disgusting to them. Then the weaker humans decided that eating animal meat was better than no meat at all and started eating other animals too, but to this day they are disgusted at the idea of eating most animals. Humans still don't eat the majority of animals. The reason they started eating animals is simple: it was easier than hunting and eating humans. Why did humans started eating human brains? I think it was one of the many accidental discoveries of human civilization: humans were killers before they were cannibals, and one day they realized that they could eat the brains of their victims, and that the brains were highly nutritional. My feeling is that "hunting" is just an evolution of "fighting". Humans are equipped with an uncanny ability and passion to kill fellow humans. It is an art in itself. Humans were killers from the very beginning. They became cannibals much later, more or less by accident. They became meat eaters and animal hunters even later.
Fish is a different story. Humans can and do eat raw fish with a relatively high chance of not getting sick. And there is evidence that a fish diet is not easily replaced by a vegetarian diet.
I would be happy to list counter-arguments that meat is a natural diet for humans. So far i haven't found one. Many readers send me links to "scientific" studies but either the studies were done by meat companies (...) or they end up proving the opposite. For example, the often repeated story that chimps eat meat: in fact, chimps "hunt" but it does not seem that the reason they hunt animals is nutritional, and in any case meat represents at most 5% of their diet. There seems to be consensus that chimps (our closest relatives) use meat as status symbol to win sexual favors and establish power structures (so there are also theories that humans started hunting for the same reason, and only later meat became a staple of the human diet).
(This article correctly points out that "In the early 1960's...it was thought that chimpanzees were strictly vegetarian." This was not only the opinion of western experts who check out a chimp for a few minutes, but the opinion of all the African natives who had lived with chimps for millennia. How could something so visible have escaped thousands of generations of African observers? is it possible that chimps started eating meat only since the 1960s? is it possible that chimps learned it from their human cousins?)
Note that this is *not* an argument in favor of being vegetarian. The point is that eating meat is unnatural. So are many other things that we do, from wearing clothes to jailing people who steal food, from going to work in offices to banning sex with underage girls. Civilizations themselves are "unnatural". Just about everything we do in a day is "unnatural". I am just saying that eating meat is "unnatural" in that our body was not programmed to eat meat. There are many wildly "unnatural" things we do. We can argue forever if it is good or bad to do unnatural things (first we should define what is "good" and what is "bad", which would be another long discussion). From the viewpoint of a diet, maybe the "good/bad" and "natural/unnatural" discussions are a bit easier: your mind may have adapted to a lot of changes, but your stomach is still fundamentally the stomach of a million years ago.
For what it's worth, here is an article from the Economist