I have heard that the human body is naturally made for a vegetarian diet. Does the shape of our teeth, length of our intestines indicate a natural make towards a vegetarian diet? Are there any other human traits suggesting the same or otherwise?
This is hard to answer objectively. Especially when dealing with the "natural" concept.
According to this Nature article, human evolution was significantly helped by eating meat. More specifically, this helped sustaining the more and more energy required by larger brain and body:
The origins of the genus Homo are murky, but by H. erectus, bigger brains and bodies had evolved that, along with larger foraging ranges, would have increased the daily energetic requirements of hominins1,2
Yet H. erectus differs from earlier hominins in having relatively smaller teeth, reduced chewing muscles, weaker maximum bite force capabilities, and a relatively smaller gut3–5. This paradoxical combination of increased energy demands along with decreased masticatory and digestive capacities is hypothesized to have been made possible by adding meat to the diet6–8, by mechanically processing food using stone too.
Another article gives some arguments in favor of omnivores humans:
Argument against evolving from vegetarian animals
Our closest relatives among the apes are the chimpanzees (i.e., anatomically, behaviorally, genetically, and evolutionarily), who frequently kill and eat other mammals (including other primates).
Relative number and distribution of cell types, as well as structural specializations, are more important than overall length of the intestine to determining a typical diet.
Nearly all plant eaters have fermenting vats (enlarged chambers where foods sits and microbes attack it). [...] Humans have no such specializations.
Although evidence on the structure and function of human hands and jaws, behavior, and evolutionary history also either support an omnivorous diet or fail to support strict vegetarianism, the best evidence comes from our teeth.
On the other side, a PETA article argues in favor of vegetarian diet using arguments related to human physiology (teeth, jaws, and nails, stomach acidity, intestinal length), health effects (fat and cholesterol increase, excess protein).
According to biologists and anthropologists who study our anatomy and our evolutionary history, humans are herbivores who are not well suited to eating meat. Humans lack both the physical characteristics of carnivores and the instinct that drives them to kill animals and devour their raw carcasses.
This article compiles a list of sources related to why humans are "natural vegetarians":
Going through the comments of some of my recent posts, I noticed the frequently stated notion that eating meat was an essential step in human evolution. While this notion may comfort the meat industry, it’s simply not true, scientifically.
Clearly, there is no consensus regarding this issue. Personally, I think that arguments in favor of omnivorous human are more convincing than those in favor of vegetarian human (arguments presented in a more scientific way).
Regardless of which option one chooses (being a vegetarian or omnivorous), nowadays we have more choices than humans could have in the prehistoric times:
we have access to various sources of proteins, vitamin supplements (particularly to B12) to allow us not to eat meat, if we do not want to
As a conclusion, I think it does not really matter if we are "natural vegetarian" because we have the change of choosing the diet and still stay healthy.
In addition to Alexei's answer, it's worth to cite a few more facts.
Humans are most closely related to the great apes. Humans and apes are remarkably similar biologically. In the wild, apes and monkeys consume diets composed largely of plant foods, primarily the fruits and leaves of tropical forest trees and vines. Considerable evidence indicates that the ancestral line giving rise to humans (Homo spp.) was likewise strongly herbivorous (plant-eating).
In another study the same researcher lists more details about the diets of wild anthropoids, resuming:
both monkeys and apes feed primarily on plant foods, eating moderate to trace amounts of animal source foods (ASF), generally insects.
We might so summarize that humans are predisposed to a diet strongly based on plant foods, being able to process and digest only small amounts of animal source foods. (We might also talk about ethics and human ability to choose, but this would be off-topic here.)
It's also interesting to have a look at other considerations about present diet: the foods we eat today are very different frome the ones we ate in the past. In the brilliant article "Origins and evolution of the Western diet: health implications for the 21st century" Cordain et al. analyze the changes that happened in foods and diet in only 150 years:
food staples and food-processing procedures introduced during the Neolithic and Industrial Periods have fundamentally altered 7 crucial nutritional characteristics of ancestral hominin diets: 1) glycemic load, 2) fatty acid composition, 3) macronutrient composition, 4) micronutrient density, 5) acid-base balance, 6) sodium-potassium ratio, and 7) fiber content. The evolutionary collision of our ancient genome with the nutritional qualities of recently introduced foods may underlie many of the chronic diseases of Western civilization.
While points 3 to 7 address to give value to plant-based diet - while not excluding the presence of small amounts of animal source foods - it's the second point, "fatty acid composition", that raises more interest and concern about the choice of consuming meat:
The dominant (>50% fat energy) fatty acids in the fat storage depots (adipocytes) of wild mammals are saturated fatty acids (SFAs), whereas the dominant fatty acids in muscle and all other organ tissues are polyunsaturated fatty acids (PUFAs) and monounsaturated fatty acids (MUFAs) (11). Because subcutaneous and abdominal body fat stores are depleted during most of the year in wild animals, PUFAs and MUFAs ordinarily constitute most of the total carcass fat (11).Because of the seasonal cyclic depletion of SFAs and enrichment of PUFAs and MUFAs, a year-round dietary intake of high amounts of SFAs would have not been possible for preagricultural hominins preying on wild mammals. Even with selective butchering by hominins, in which much of the lean muscle meat is discarded, MUFAs and PUFAs constitute the greatest percentage (>50% of energy as fat) of edible fatty acids in the carcass of wild mammals throughout most of the year.
In other words domesticated animals (and even more the industrially-raised ones) are much more rich in saturated fats than wild ancestors, thus suggesting they're unfit for human consumption.
We might also analyze more in depth wether the conditions where the animals are raised (use of antibiotics, growth-hormones, etc.) make them suitable for consumption but this goes far beyond the scope of your question.
Humans are omnivorous, meaning that we can consume a wide variety of foods. Other omnivorous mammals include dogs, pigs, badgers, bears, hedgehogs, opossums, skunks, sloths, squirrels, raccoons, chipmunks, mice, and rats. Hominidae (i.e. great apes, including humans) are also omnivores.
Omnivorous animals can survive on a diet of meat (including fish, insects, and eggs), a plant-based diet, or a combination of both. Omnivorous animals can adapt to circumstances and survive on whichever foods are available.
At different times in History, humans have adapted to varied diets, including the evolution of special genes that allow for digestion of lactose as adults, but at the same time we've lost the ability to digest large quantities of raw meat. Thanks to farming, and unlike other animals, humans can choose what to eat.