You might say that paladins are the guardians at the gates of hell -- they fight evil wherever it penetrates into their world and they take the fight to the evil's source in the hope of quenching it forever. Although they focus on guarding their people from undead and demonic forces on the rise, paladins actually stand against evil everywhere, including the evil in their own hearts.
Being a paladin means that you have a relationship of some sort with the Holy Light, that mysterious force of goodness and faith that flows to some degree within all living beings with positive intentions. Most paladins (and many priests) believe that when you do something that you believe to be good, the power of the Light increases in you and your connection to the rest of creation is strengthened, whereas doing something evil (such as acts of greed, despair, or vengeance) will darken the universe and weaken your connection to it. Whether this belief system is a religion or a philosophy is open to interpretation, and seems to depend in some part upon which race you are.
There are three sorts of paladins in World of Warcraft, aligned with the humans, the draenei, and the blood elves. All of these share certain similarities, but each has its own differences as well.
Paladins among the humans (and dwarves too) are not an ancient order by any means. Originally there were just the Clerics of Northshire -- priests who were not well equipped for battle and suffered great losses during the First War, (about 25 years before the present). Afterwards, their leader, Archbishop Alonsus Faol, recruited knights and clerics both to teach one another and become great warriors skilled in both warfare and healing. These new warriors of the Light were called Knights of the Silver Hand, based in Lordaeron, and they were very helpful in winning the Second War six years after the start of the first. Uther the Lightbringer, probably the most famous paladin, was the first human paladin to show his abilities on the battlefield, and was the instructor of Arthas Menethil.
But Arthas betrayed the Order of the Silver Hand during the Third War after his lust for vengeance so consumed his mind that he lost his morality and joined the very legions of undead he once sought to destroy. With his necromantic powers as a death knight, he killed many paladins, including his mentor, Uther the Lightbringer, and left the Silver Hand in disarray, unable to defend against his legions of undead that swept over Lordaeron. Those paladins that remained either retreated into fanaticism (as with the members of the Scarlet Crusade), or rallied with other noble souls (especially the dwarves) through Light-inspired initiatives such as the Argent Dawn.
Up until the outset of Wrath of the Lich King, human and dwarven paladins have been struggling to pull themselves together after the great losses at the hands of Arthas. Now, however, Tirion Fordring has taken up the mantle of leadership and made good of his vow to restore the Order of the Silver Hand to its former glory and gathered all paladins together into the new "Argent Crusade."
As a human or dwarven paladin, you have lived through many of these experiences, each of which would have affected you profoundly. If your character is older, you may have started out as a warrior or cleric before paladins even existed, but a young person (about Arthas' age) would have grown up in the golden years at the height of paladin glory.
Likewise, if you have studied a lot of the theory about the Holy Light, you may consider it more of a philosophy and way of life rather than a religion. In practice, however, most humans tend to use religious phrases and customs when talking about the Light, as seen in their prayers, their construction of cathedrals, altars and so on. For many, "Light" seems to be the Azerothian name for "God." What your character believes is up to you.
The paladin tradition of the draenei goes back many thousands of years, to the time 25,000 years ago, when Velen first encountered the naaru and accepted their assistance in escaping from the eredar. Ever since that exile, the draenei have been students of the naaru, and have come to revere the Light very deeply, even if they are not paladins or priests.
The draenei seem to hold a more religious view of the Light, calling upon it either as a saint might invoke God, or a Jedi might invoke the Force. And, although the exact nature of the naaru isn't entirely clear, they are clearly representatives of the Light to the draenei, wellsprings from which the energies of the Light flow.
To be a draenei paladin is to belong to one of the longest and most prestigious traditions in the history of your race, the very core of who you are. Meeting the humans after arriving on Azeroth would have been a miraculous confirmation of the Light's power to such a character, and would have given him or her great hope in the face of the draenei's 25 millennia of struggle against the Burning Legion.
We have discussed the story of the blood elves and their relationship to the Light at length, but their order of "Blood Knights" deserves another special mention here. As you know, the blood elves originally sought to "steal" the power of the Light and bend it to their will by enslaving one of the naaru, but later on it turned out that their captive naaru had chosen this path for himself, knowing full well what the future would bring, which, in a sense meant that the powers of the Light was a gift to the blood knights, not a stolen energy they had taken away on their own. When that same captured naaru was transformed into the essential energy that reignited the Sunwell, the power of the Light flowed out to all blood elves everywhere, satisfying their need for magic and planting the seed of the rebirth of their civilization.
Needless to say, blood knights who had once prided themselves on their ability to steal and manipulate their way to power with the Light must now face the humbling fact that this power had been a gift all along. Indeed, the very creature they once tortured for this power has recently sacrificed himself for their sake, and now makes the Light available to them in abundant measure whenever they wish to draw upon it. Undoubtedly, this should cause a crisis of faith in many blood knights, with numerous possible outcomes, although I suspect most would seek redemption and forgiveness for their former arrogance.