Elune is the only official "god" in the Warcraft canon, outside of the Old Gods, which the general consensus seems to consider more of a title than an actual descriptor. While many races worship the Light, the Light is generally considered more of a cosmic principle than a singular conscious entity. The Draenei revere the Naaru, but they don't strictly worship them, as the Naaru are more akin to Angels in the Judeo-Christian-Islamic tradition than actual deities. The Trolls do worship Loa, and it's possible that these beings fit your classic definition of "gods" but they are not called as such (I've always assumed that Troll Loa and Night Elf Ancients - and Pandaren Celestials, which Blizzard has explicitly connected to the Ancients - are all effectively the same sort of being.)
But Elune is considered a straight-up Goddess by the Night Elves.
The closest thing to your Classical gods (classical meaning Greco-Roman, which are probably the biggest influence on Western mythological culture outside of the imported Abrahamic monotheism - though the Norse gods are also a big influence) are the Titans (Titans of course being an imported term from Greek mythology.) The nature of the Titans has always been mysterious, as it seems that they are somehow both divinely powerful, but also have a scientific/technological basis for their powers - they are your classic "sufficiently advanced" society. (Arthur C. Clarke, one of the biggest 20th Century sci-fi writers, suggested that any sufficiently advanced technology is indistinguishable from magic.)
Elune has always kind of been a mystery in Warcraft, because she seems so different from the objects of faith for every other culture. While the Night Elves revere or even worship the Ancients, they still have a fundamentally monotheistic religion. Cenarius, for instance, is revered in part because he is the son of Elune. And because Cenarius most definitely exists, we can be fairly confident that Elune does as well.
Some have theorized that Elune is, in fact, a Naaru (Blizzard kind of cheekily suggested that Velen had come to visit Darnassus and remarked that he thought she might be, which did not sit well with Tyrande.) It would make sense, as she's this vastly powerful benevolent entity that empowers her worshippers with Light (even if it's Moonlight instead of Sunlight.)
But some new details emerging from the Legion Alpha might give us an alternative explanation of just what Elune is.
As we quest through the Broken Isles, we'll be collecting Pillars of Creation, powerful Titan artifacts that played a key role in the creation of Azeroth. We'll be trying to use these Pillars to close the portal the Legion will have opened at the Tomb of Sargeras. Each pillar seems to be linked to a different Titan. For example, it seems that the one in Highmountain, held by the Drogbar chieftain, is the Hammer of Khaz'goroth.
And one of the Pillars is called the Tears of Elune.
Does this mean that Elune is a Titan?
The Titans are certainly the closest thing we have to your classical "gods." But we've never heard Elune mentioned as a member of the Pantheon. The Titans were originally presented as a race, over whom the Pantheon were the leaders, so it's possible that Elune was simply not a member of the Pantheon.
But perhaps there's another explanation.
The Pantheon is composed of Aman'thul, Eonar, Norgannon, Khaz'goroth, Aggramar, Golganneth, and formerly Sargeras (Aggramar took up his old position.) It's perhaps odd that Eonar is the only female member of the Pantheon (though I might just chalk that up to careless writing.) We do know that in the early days of Azeroth, when the Titans came to battle the Old Gods and cast down the Black Empire, one Old God died and one Titan fell. We learned that the Old God Y'Shaarj was the one who was killed, thanks to the work of Watcher Ra (Ra-Den) and the pre-Curse of Flesh Mogu. But we never really found out which Titan had fallen, and whether "fallen" meant slain or something else.
The Well of Eternity was clearly one of the most important features of Azeroth. Mere off-shoots of it at Mount Hyjal, Quel'danas, and the Vale of Eternal Blossoms have all been incredibly powerful sources of magic and power. The first Night Elves were most likely a group of Dark Trolls (a subrace that hardly exists anymore) who settled on its shores and were transformed, notably into more humanoid/Titanic morphologies, by the magics there (a similar thing happened to Murlocs who were transformed into Jinyu by the waters of the Vale of Eternal Blossoms.)
The Night Elves then came to worship Elune as their primary god, certainly revering other nature spirits, but with Elune as their most exalted deity.
Jumping well ahead, when Wrathion ate the heart of Lei Shen, he had some sort of vision that allowed a strange voice to speak through him, speaking about "rebuilding the Final Titan."
Could it be that the Titan who fell in the war with the Old Gods was that Final Titan? And that having fallen, it had to be rebuilt? The Titans are beings of metal and stone, and as such, perhaps the very concept of "death" is not really relevant to them.
And as beings of such immense power, it's possible that their blood could be an incredibly powerful source of magic.
So here's a proposition: Perhaps Elune was that Titan who fell in battle with the Old Gods. As she was struck down, her blood seeped into the world and became the Well of Eternity. She was not killed, exactly, and she remained a conscious force with some small portion of her original power. To keep her safe until she could be rehabilitated, she was placed in a large protective structure that was set into orbit around Azeroth, becoming Azeroth's white moon.
From there, she has been able to use some of her small remnant of power to aid Azeroth, empowering the Elves who had been transformed by her blood.
And because the Titans are the embodiment of all that Sargeras wishes to tear down, he has been trying to conquer Azeroth in order either to prevent Elune from being restored, or to corrupt her to be another Dark Titan.
And the Avatar of Sargeras was left within a Temple to Elune.