Long long ago, human beings all around the world (of Earth, not Warcraft) investigated different ways of describing how the world around them worked. Many different cultures found that the materials they encountered seemed divided into four or five separate elements, each with its own properties: earth, fire, water, and air. Space, "void," or "aether" was often noted as the fifth element, or, as in the case of China, the understanding of these elements looked a lot different but in the end produced a similar sort of system.
In Azeroth, however, these ideas about the elements never got swallowed up by modern science and the periodic table of elements. They turned out to be real forces in the world, each with its own set of elemental spirits, which people could communicate and cooperate with.
Shamans are the masters of this magical task, charged with helping to maintain the balance of nature in a very different way from druids. While druids are focused more on nature as a system of energy, life, and growth, shamans focus more on the spirits of the land, flames, waters and skies as they all interact with one another. They gain great wisdom by learning of the different characteristics of these elements, and in turn bring this wisdom to the people they serve.
The spirits of the elements exist all across the Warcraft universe, in more or less the same state on all worlds with life on them. To most mortal creatures these spirits are invisible, however -- it is only when they choose to share their wisdom with a particular race that a people may come to know of their existence. On Draenor, it was the orcs (and later the draenei and the broken) who came to know of the elemental spirits and the path of the shaman, while on Azeroth, the tauren were the only ones to discover them (the trolls learned about shamanism from the orcs later on). Each of these races practices the same basic form of shamanism, since the spirits which teach them are the same.
A shaman does not cast spells in the same way that most other magic users do. The power of the elements is not like modern human electricity which they can plug into and then channel through their body or their mind. Rather, shamans call upon the spirits when they seek to use this power, asking permission to use it and channel it through themselves. The totems a shaman creates represent this relationship, and serve as a focal point of their communion with the spirits. So in a sense, each shaman must humbly remember that his power is not his own, he borrows it from the spirits of the world, and uses it at their good pleasure. Each type of spirit represents a force of nature at work in the world, all existing in balance with one another.
The spirit of Earth is the force of stability and trust. Its nurturing quality allows crops to grow, while its strength and firmness allows it to quake and destroy its enemies. Its voice is steady, slow, warm and persistent.
The spirit of Fire is the force of destruction and illumination. For unpredictable reasons of its own, it may alternately wipe away all the withered death of the world, making way for new life, or when it moderates its hunger and passion, it may serve as a nurturing force too, as in a candle or fireplace. Sometimes Fire may even grant shamans visions of the future or far distant places.
The spirit of Water is the force of healing and adaptability. It adjusts to any situation in an appropriate manner, whether with force or with infinite patience, no matter how severe a calamity it may be. It is the voice of wisdom, clarity and mirth.
The spirit of Air is the force of mystery and communication. It may take the form of a gentle breeze or a violent storm, for reasons mortals may not comprehend. Yet at the same time, its gentle voice often whispers in the ears of the shaman, a constant companion wherever he or she may go.
Different sorts of shamans may have different relationships with the spirits of nature. They may feel a special affinity for one over the other, and may even know many individual spirits by name. They might even have a kind of friendship with one another, although not a relationship of equals as mortal people might share -- the spirit would definitely be in a place of reverence or worship. A shaman is also more likely than most other classes to have special rituals and practices that are overtly religious in nature; other classes may also have strong beliefs, but not many others are able to commune with spiritual entities that actually talk back!
Each race that believes in shamanism follows not only the spirits of the world but also its own ancestors. The shamans of the past guide their people and help them to understand the ways of the spirits.
Although the draenei themselves are an ancient race, they are the newest to take up the knowledge of the spirits. A draenei shaman does not have many shamanistic ancestor spirits to act as guides, but the living instructors such as Nobundo teach shamanic knowledge accurately.
The tauren, on the other hand, have communicated with their ancestors for generations. These ancestors hold such esteem in tauren culture that each individual tauren learns to recite his or her own lineage stretching back many generations. Shamanism is the default way of life of all tauren, even if they are not themselves shamans.
Like the draenei, trolls are somewhat new to the practice of shamanism, although in this case the shamanic belief system supplants rather than supplements their previous religion. Trolls who have learned the ways of the shaman from the orcs can turn away from the dark voodoo rituals and beliefs which they followed in the past, although to some degree they may be unwilling or unable to totally separate the two. A troll witch doctor may become a shaman, and try to blend voodoo and shamanism into his own particular mixture.
People who roleplay orcish shamans have a lot of special opportunities that other races do not have. An elderly orc may have learned the ways of the shaman as a youth on Draenor (now known as Outland), back before his people were corrupted by the demonic power of the Burning Legion. Such a character may have even followed the demonic path for a while and become a warlock, as Drek'Thar did, only to return to the path of shamanism later on in life, filled with the deepest regret for their actions while under the influence of demons.
A younger orc shaman would have an entirely different story, however. Like Thrall, his or her path to shamanism might have been a path of discovery and amazement. He or she might have grown up in Azerothian internment camps and learned of shamanism while Thrall was reuniting the orcish tribes into the new Horde. Surely for this kind of character, Thrall would be the ultimate hero and role model, and shamanism would be a way of reclaiming everything that was good about your people, arising out of the depths of lethargy into the bright field of meaningful action.
Tangential note: "Shaman" or "shamans"?
There is some disagreement as to which plural form of "shaman" is correct. The English language dictionary, on the one hand, says that "shamans" is the correct word, while Blizzard has stated that "shaman" is the correct way. However, Blizzard documents themselves are notoriously inconsistent, and sometimes you can even find both spellings on the same Blizzard webpage.
The official All the World's a Stage pronouncement on the correct spelling of the "shaman" plural form is this: You must spell it "shaman" or "shamans," depending on which form you like best! Also, arguing about it with other people does no good whatsoever, and such arguments started up by strongly opinionated individuals are best concluded by admitting that the other person is right.