Humans are the 'original' race of the Destiny universe. They're the first race we come upon because the entirety of the game is based upon the survival of humanity. Humans are still plentiful in this future/apocalyptic landscape in time, though the numbers were drastically dropped from the original billions of humans available due to the Darkness interfering with the humans and their escape from danger.
Exos were originally created by the Clovis Bray corporation during the Golden Age brought on by the Traveler. Exos were made for immortality, but due to real humans having to be used to 'test' the functions of the exos the corporation wasn't able to further their research until a deal was made, allowing the corporation to use prisoners as test subjects for the exokind. Exos are a completely robotic 'race' that have the mind of the human inside a brain of sorts, like a hardrive used for an entire human personality.
Awoken were born from the end of the Golden Age, a mix between light and dark colliding on the edge of the solar system. The original Awoken born from this clashing of two equally powerful primordial powers were sent to their own dimension, per se, though a small portion of the Awoken wished to leave their immortality to return to their origins. Once the Awoken happened to come by the way at making it home, they settled in the reef and what was left of the population of Earth.
Fallen are a humanoid race with four arms, and they were the first to be welcomed by the Traveler. They were once a graceful race that were strong and vigorous, feeding off of a plentiful source of Ether to keep their strength intact. When the Travaler finally left them, the source of their life, they followed after it. Without their infinite source of Ether, they would dwindle out and turn into a race of scavengers and pirates. Now the only source of Ether for the Fallen, or Eliksni, are the servitors that frequently travel with groups of Fallen.
Hive are power-hungry creatures that go by something known as the Sword Logic, which is the idealogy that murder would allow someone to gain more power. This idealogy was introduced to them by beings known as the Wyrm gods, which were the ones feeding off of the death and granting power to the Hive. The Hive were once ruled over by a being so powerful among the Hive that he could 'take' things from their own will and turn them into slaves of some sort. Oryx was the 'god among men' when it came to the Hive, though he was slain.
Vex are a liquid-type race that are highly advanced in the field of technology. Vex are able to use the functions of time as a playground, being able to manipulate the past, present, and future in areas that they have total control over, such as the Vault of Glass where the Vex 'God', Atheon, resided. The Vex control a various amount of planets throughout space, and are though to control a large portion of space due to their pristine advancement in technology of all sorts.
Cabal are just as power-hungry as the Hive, though they do not have help from some sort of godly beings, they merely destroy and take over as many areas as they can. They are a highly militarized race they take over whatever they desire, though their only real competitors have been the revived guardians produced by the Traveler. The Cabal are a widely untapped race due to the hold they have on so many areas, though not in our solar system.
Taken originated from Oryx, the "Taken King". The Taken are made up of all the other different races of darkness, Fallen, Hive, Vex, and Cabal. Some of the Taken earn abilities that can be fairly annoying to deal with. The Taken psions are able to duplicate almost continuously if not taken care of. Taken phalanxes are able to practically throw their opponents away from them. Last but not least, some are even able to protect their fellow allies which make killing them much more tedious. With Oryx dead the Taken are thought to be controlled by Savathun, Oryx's powerful sister.
Scourn, to be put delicately, are dead Fallen. A Fallen that created a source of corrupted Ether, known as The Fanatic, is able to tap into immortality by using this corrupt Ether. The corrupt Ether kills any fallen it enters, though it grants them immortality through resurrection. The Scourn are a race of Fallen that would be considered 'rejects' from the rest of the Fallen community.