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There is not currently any in-game lore for humans in Cataclysm, but humans have always been a race capable of anything. Ashley Blank is your first trainer.
As descendants of the Earthen, Dwarves are very much a part of the mountains. Hunting is pretty steeped in tradition and dwarves enjoy learning the ways of the land and tracking a animals such as bear through the mountains. 
Night elves possibly more at home in the woods then any other race. Able to shoot a bow with unerring accuracy, they hold a deep respect for their homelands and the lands beyond. 
Many of the greatest hunters are a part of The Sentinels, the elite army changed with defending the night elf lands. The night elf hunters can often be found with their Nightsaber companions.
Though not really possessing the looks of a hunter, the draenei once hunted beasts (mostly fel boars) on Draenor, now called Outland. Now exiled on Azeroth, they discover the environment that draenei never saw, unfamiliar beasts and plants, and now rediscovering themselves as masters of the wild. Under some minor supervision from the shaman Farseer Nobundo, they tame the world little by little.
There is not currently any in-game lore for worgens in Cataclysm, but it's most likely the same as the humans. Your first trainer is Huntsman Blake.
A blood elf hunter is essentially a Farstrider having passed the techniques down for generations. The elven rangers are some of the most sophisticated archers in the world. While most of the blood elves prefer to be Blood Knights, many still choose the old ways of a Farstrider, and because of the natural content and affinity a Farstrider has, the blood elven hunters are slightly less arrogant than most other blood elves.
Orcs of the Horde first learned the ways of the hunter from forest trolls on Lordaeron. Many orcs find the path appealing due to their increasing closeness with the wilds.
Orcs were hunters back on Draenor, but this refers to the profession rather than the class, not all orcs have the hunter class. See below for more information.
Tauren are natural hunters. Tauren have been masters of the hunt since the dawn of the world. Their culture also revolves around the great hunt, signifying the importance of the hunters (or at least hunting).
Jungle trolls are natural hunters.
The mighty sons and daughters of Nath are natural hunters, having spent long years in the wilderness. They have a natural rapport with wild animals and even those who are not hunters often enter battle with loyal, trained companions at their sides. They seek to discover the essence of combat through observing and mimicking wild beasts. Most half-ogre also become melee hunters; those who don't use massive composite longbows - something to which they can apply their great strength. The iconic half-ogre hunter emerges from the woods or desert on the eve of battle, silently taking their place next to their allies in the Horde. A great bear, lion or wolf remains by their side, looking docile enough, but growling when others get close. The hunter hefts their axes and looks to the horizon. They breathe deep, savoring the smell of the land and preparing themselves for the conflict to come.
Forest trolls on Lordaeron were hunters. They taught the orcs to hunt.
The high elf rangers were hunter-like, they inherited the skills from the ancient night elf rangers. There are some examples of high elven hunters like Lor'themar Theron.
Blood elves also inherited the hunter profession.
Hunter refers to two fields, hunter class, and the hunter profession. They do not mean the same thing.
Practically anyone from any class can be a hunter (that is having the profession). A hunter with the profession is one that hunts for food for their own survival or survival of their group. It is a survival skill rather than a fighting style (although as in real life it could be used in defense if need be).
However very few actually take on the hunter class (which is a very specific combat class). Someone with the hunter class is someone with a very specialized fighting style, almost the equivalent to an elven ranger, but with its own specialized fighting abilities.
For example, note the reference to orc hunters above;
When it is said that all orcs were hunters back on Draenor this does not mean that every single orc has the hunter class. On the contrary it refers to the fact that most if not all could hunt game for their clans. This usage of the term refers to profession or hobby of hunting rather than the class.
However, when it is said that trolls taught the orcs the way of the hunter, it is referring to how the trolls taught the orcs the ways of their specialized hunter class.
This being said, those of the hunter class are in-touch with nature and spend much time in the wild, often possessing many of same skills and survival know-how of hunters-by-profession, and the one lends to the other well.