So, I've played through the story a couple times, cuz hey, I actually liked it.
I've typed up this little wrap-up of the races, their apparent motives, and what we do and don't know. Originally, this was a response to one of those halfwit haters who can't understand the concept of progressive storytelling, but I thought I'd just put it here for all to see, not just in a reply to jackass troll #4368752. Here we go...
The Cabal are simply brutish imperialists, endlessly advancing and conquering worlds. The Grimoire says, "You could follow a trail of broken planets back to the Cabal homeworld (sic)". It seems that they don't much care for anything more than conquering for conquering's sake. They're just a warlike race, from what I know.
The Fallen are scavengers. They go where there's loot to be had, from either abandoned warzones and cities, or by preying on what they think are weaker races. They do seem to be fighting the Hive a lot. This might be due to the Hive just being murderous of other species, or the Fallen trying to claim their land/treasures.
The Hive are a more bestial, primal race. They have colonies like insects, and there's a lot of dimorphism. We don't know if they're all one species, or a collective of symbiotic species. This is left as a mystery, since they're the more horrific of the races. This is a common storytelling trope - not revealing too much info on the scary monsters - since people tend to be more afraid of things they don't understand. I'm sure we'll learn a lot more about them in The Dark Below. All we really know about them at this point is that they're closely related to The Darkness, and those wizards are dicks.
The Vex is not really a race, but rather one consciousness spread across space and time. They work toward an end that is likely unfathomable to our characters, since we're just high-tech humans, and mechanical hive-minds probably have different wants or needs (if they even have wants or needs). We do know that they're trying to build a network across the solar system, perhaps even across the universe. The Vex also seem to be an imperial race, in regard to their attempts toward this goal.
The Vex found the Darkness in The Black Garden (or maybe they built the garden around it), and saw it as a way to become more powerful, thus they supported it and protected it, in hopes for some of its power. Perhaps the Vex were even trying to control it. The Traveler is the enemy of The Darkness, which is why The Vex are at odds with it and humanity.
The Darkness wants to extinguish all light and life from the universe. This is obviously bad news for all the races, which gives each race reason to fight against the Vex. That's why you see the Fallen and Cabal fighting them on Earth, Venus, and Mars.
At the end of the story, we destroy the fragment of Darkness that had enveloped the shard of The Traveler, thus pushing back the Darkness (for the time being) and restoring The Traveler. And now the real fight begins. We've won the battle, but not the war. The Traveler is awoken, and The Vex and Hive will be fighting harder than ever to destroy it.
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My theory: the traveller is one of its race which is all dead except for it and the one that betrayed and killed its race: the darkness. Long ago the travelers where a mighty race that brought glory to all realms uplifting them to a higher standard. You see our traveller, which we will call "light" was the youngest of these beings, it ventured into a new galaxy where it found us. But in the travelers realm a dark threat grew. A traveller named oryx sought to rule the universe and to this end created a virus made to kill all travelers, oryx vaccinated himself against it but it twisted his body, resulting in his differing look. The virus wiped out all but one traveller in the galaxy. That one traveller was named ether, and he uplifted a factious race which we call the fallen. Oryx needed an army to conquer the fallen so he transformed the species he once guarded into the hive. He also used the remains of his dead brothers and sisters to create the vex. To this day the hive and vex worship him as a god. Oryx then sought to conquer ether and the fallen but was repulsed. Seeking an edge he aligned himself with the empires of a race known as the cabal. He then launched an attack that broke the fallen. Ether sacrificed itself to become the servitors and primes. Oryx tricked the fallen though, corrupting ethers shards to trick the fallen into thinking that the only way to revive ether was to take parts of our traveller, light. And so oryx and his hordes attacked earth, oryx was not able to wipe us out bit did steal a shard of light and used it to keep light from healing. Oryx then set out to conquer the galaxy of the travelers. He now returns to take our galaxy, crush his brother light, and take the universe as his own. That is where we are today.