Priests in the World of Warcraft are a single class that incorporates a wide variety of characters. They are best known for casting spells that call forth the power of the Holy Light, but the priest using these spells in the game mechanics doesn't necessarily have much connection to the Light as such -- rather they have a connection with their own religion which grants them similar effects to those of the Light.
When WoW was being developed, Blizzard realized that night elves and trolls, for instance, would not follow the Light in the same way humans and dwarves do, so they tried to represent a bit of this diversity through race-specific spells. It didn't work out, though -- some were too powerful, while others weren't worth reading about, much less putting on one's action bar. The end result was that they made some of these spells universally available to all priests, and completely removed the rest. Here the lore had to surrender to the game mechanics in order to provide the best game balance.
In roleplaying, however, there is a lot of room for players of different races to behave differently, and draw their powers from totally different sources. Greater Heal, for instance, could come either from the Light or the power of Elune. A Shadowfiend could either be a spawn of the Forgotten Shadow, or a dark trollish voodoo spirit. If you are roleplaying a priest, the only thing that really matters is that your character have some sort of faith or profound belief, which could serve as the source of their divine magical power. A priest's magic revolves around his or her strong beliefs and ideas -- but what those beliefs are is entirely up to you.
The two most popular sources of priestly divine magic (which is quite different from the arcane magic of mages), are the Light and the Shadow. Humans, dwarves, draenei, and to a certain extent blood elves as well, are all known for drawing on their faith in the Light to heal and support their compatriots in battle. Humans and dwarves are members of the Church of the Holy Light, based in Stormwind; the draenei are universally faithful to the extent that they don't even have a separate "Church" organization of any kind; and the blood elves are rather confused about what they believe right now -- many have regained their faith in the Light, but some still believe primarily in power and control as their ultimate religion.
We talked about the Light a bit in the article about Paladins, who draw upon the Light exclusively. Those priests who believe in the Light also draw upon it a great deal in battle just like paladins do. Priests of the Light, however, have a special duty to guide the people around them in social and spiritual matters as much as (or even more than) protecting them in warfare. Common people tend to expect priests of the Light to be upright and moral role-models, even if they do not believe in the Light themselves (although many other adventurers may be much more skeptical after getting out and about a bit).
Shadow is a much less understood form of magic. If one merely looked at its use as a type of magic in the WoW game mechanics, we could see it is used by demons, undead, and any sort of generally evil enemy. Yet, even the most Light-fearing, good-hearted of priests must draw upon shadow spells (such as Shadow Word: Pain) in order to do any significant damage to an enemy. A priest leveling up will get into a lot of trouble using Smite to the exclusion of all shadow spells. One could simply ignore the fact that one's priest uses these shadow spells, but it makes more sense to simply find another way of incorporating it into your character's belief system.
Although the Shadow used by priests is not clearly described in any official lore source I could find, it appears to be something different from the necromantic magic of the Scourge and demonic magic of the Burning Legion. Many roleplayers understand it a sort of counterpart to the Light -- not a kind of evil energy in itself (the way that fel magic is) so much as a force of natural decay and death within the overall healthy cycle of life in the world. It is the yin to the Light's yang, and together, both sides provide the balance that is life itself. The Forsaken in particular embrace this Shadow element through their "Cult of the Forgotten Shadow" religion without ever going to the extremes of the Lich King or the Burning Legion. The Forsaken essentially just realigned the teachings of the Church of the Holy Light to fit their worldview as underdogs seeking out more power for themselves in a world that has forsaken them.
[Edit: The Light itself isn't a as big a part of the Forsaken's faith anymore (they claim it has forsaken them, naturally), but they still acknowledge its importance. Some lore accounts say that it's impossible for Forsaken to connect to the Light anymore, but Forsaken priests still have to draw some power to heal with, so that doesn't make much sense. It's possible that some might even have reclaimed some sort of connection to it, as holy priests, or maybe they can still direct it to heal others, even if it doesn't flow through their bodies (I know that's kind of like swimming without getting wet, but what can you do?) Perhaps Forsaken are able to suck the shadow energy out of a person, creating a vaccuum in which the Light rushes in to fill up the empty space and heal it.
Also, some commenters have pointed out that Priest shadow magic could also be interpreted as a kind of psychic attack rather than actually drawing on shadow energy as described above. It makes sense that a class whose whole job revolves around faith and belief would have special control over the minds of others in some situations, perhaps even summoning up shadowfiend monsters from their nightmares. This is definitely another option for you to choose from.]
Night elf priests probably wouldn't see much of a dichotomy between forces of Light and Shadow. Since they do not follow either the Holy Light or the Forgotten Shadow, they needn't understand Holy and Shadow magics as opposites. Their goddess, Elune, is the goddess of the moon, has a light and a dark side that constantly interchange with one another in the natural course of time. She has a bond with the world, guiding its peoples, protecting them from dangers, and restoring the balance of life. A night elf may believe that Elune has nothing to do with either the Light the humans believe in, nor the Shadow of the Forsaken -- or that she is the single source for both the Light and the Shadow. To them it doesn't matter so much as the belief that there is no inherent conflict between the two.
Incidentally, the night elf priesthood was formerly made up entirely of women, and the priestly organization, "the Sisters of Elune" remains entirely female. Male night elf priests have apparently not tried to form a "Brothers of Elune" for themselves. I would pay gold to watch two night elf priests, one male and one female, argue about gender roles in their society -- as yet I've never seen any roleplayers' characters express strong opinions on the matter, even though it seems like it has been a major issue in the lore.
Trolls are the most unique race of priests, in that they hold an entirely different belief system compared to the other races of Azeroth. Their priests don't care one bit for Light, Shadow, Elune or anything else, nor do they see themselves in any way connected to the "shadow" magics of the Lich King or the Burning Legion. Instead, their magic revolves around shifting relationships with the spirits that live all over the world. Through ritual and belief, the trolls can influence these spirits for good or evil, whichever way their own heart is inclined.
A troll priest could also be called a Witch Doctor. He could perform all kinds of special rituals for other trolls, including rites to try and see into the future, as well as rites to speak with the dead or cause harm to someone else over a great distance. Evil witch doctors could even perform rites that are forbidden in the new Horde, such as the sacrifice of sentient beings, and ritual cannibalism.
In the end, whatever the absolute truth is regarding Holy and Shadow energies matters much less than your character's own beliefs about them. If you want to roleplay that your character balances the forces of life for healing and death for destruction because of his faith in the Light, that's fine. Or, you could make your priest into a mystical speaker with the dead, who uses light and shadow magic to probe the mysteries of the soul. There are a whole range of ideas that could make sense for your character.
Alex did a great article about the different religions of Azeroth, which is a great resource for any aspiring priest to find out what his beliefs should be. The important thing is that you should have strong beliefs of some sort, because your divine power comes from them. Even though, due to the necessities of game design, your priest has the same abilities as other priests, your roleplaying can create an entirely different flavor, as if you're another class altogether.