The world of Kingdom Hearts employs a variety of baseline concepts in its worldbuilding. Some of these can seem a bit dense, but there is a particular methodology to all of it. The best way to understand it is through the relationship of the Heartless and the Nobodies.
While the Heartless and Nobodies have recurring roles in nearly all the Kingdom Hearts games as antagonistic forces, their natures and origins are only briefly touched upon in the games' critical story material. By digging deeper, though, we can unravel what exactly makes these mysterious entities tick and what it is they seek.
The Three Components Of A Being
In the world of Kingdom Hearts (the overall world, that is, not just the individual "worlds"), every living, sentient creature and person is what's known as a "Somebody." Be they good, evil, or whatever else, if they're not a Heartless or a Nobody, then they're a Somebody. The being of a Somebody is made up of three primary components: the Body, the Soul, and the Heart.
The Body is a Somebody's physical being, their presence in the tangible world. The Soul is a Somebody's mind or will. The Soul is what contains things like memories, impulses, or instincts. Finally, the Heart is debatably the most important part of a Somebody. The Heart contains emotions and drives, as well as a Somebody's inner light and darkness. Every Heart, both good and evil, has positive and negative feelings, which means there is a wellspring of both light and darkness within it, respectively.
Typically, a Somebody can't exist without all three of these components. In the case of natural death, for instance, that being the destruction of a Somebody's body, their Soul and Heart are sent in tandem to the Final World, after which we can't really say what happens. However, there are exceptions to this, brought about by means both magical and mundane. It is from these means that the Heartless and Nobodies arise.
The Heartless
A Somebody's Heart can be completely consumed by darkness, either from their own inner wellspring as a result of a negative emotional overload or from an external source of darkness. Either way, when this happens, their Body and Soul are erased, leaving a disembodied Heart swimming in a pool of Darkness. This disembodied Heart takes shape in the form of a Heartless, a creature of darkness.
The Heartless are driven by an instinctive desire to spread their darkness wherever they can by stealing the Hearts of Somebodies. This is also why Heartless are instinctively drawn to wielders of the Keyblade; Keyblade wielders' Hearts shine bright with both light and darkness, serving as a beacon of sorts for hungry Heartless.
As creatures driven by the darkness, the Heartless are instinctively subservient to those with powerful wellsprings of Darkness in their Hearts like Maleficent or Xehanort. Of course, attempting to control the Heartless in this way is extremely dangerous, as losing control of your emotions will drive the Heartless into a Heart-snatching frenzy.
Heartless born from natural sources of darkness, like one's inner darkness or the Realm of Darkness, are known as Pureblood Heartless. These kinds of Heartless have pure black or purple bodies, and in the case of larger Purebloods like Invisibles or Darksides, a large heart-shaped hole in their chests. Meanwhile, Heartless born from artificial means like dark magic are known as Emblem Heartless, named after the red and black Heartless emblem stamped somewhere on their bodies. These Heartless take on stranger, more varied forms, often reminiscent of the World their Somebodies came from.
The Nobodies
As we mentioned, when a Heart falls to darkness and a Heartless is created, the Body and Soul simply fizzle out of existence. However, when a Somebody possesses an exceptionally strong Soul, driven by powerful convictions and will, the Body can continue to persist on its own as an independent entity. This is how a Nobody comes into being, an ephemeral creature that is not meant to exist.
As creatures of neither light nor darkness, Nobodies can freely traverse through both realms with little concern for their safety. They don't have Hearts that can fall to darkness, after all. Unfortunately, in spite of this perk, the transition to a Nobody is not generally a positive one. Most Nobodies undergo severe physical mutations and mental distortions, assuming bizarre, ephemeral forms and exhibiting erratic behaviors.
Unlike the Heartless, as Nobodies still possess their Souls, they are capable of sapient thought. Nobodies can plan, strategize, and coordinate with one another to get what they want. However, they do still have a base instinct they can't escape: Nobodies almost constantly long for their Hearts back, constantly adrift in an emotionless void of being.
Without their Hearts, Nobodies can't actually feel emotions, only reacting to situations based on their memories of how they would have reacted if they still had them. With enough time and emotional stimulation, it is technically possible for a Nobody to develop an entirely new Heart, as was the case with Roxas, but this is a rare occurrence.
Unusual Cases
The rules that govern Somebodies, Heartless, and Nobodies are generally iron-clad, but there have been some exceptions over the years. One of the most prominent of these exceptions is the many incarnations of Xehanort. Due to possessing both an extraordinarily strong will and a natural affinity for the darkness, both Xehanort's Heartless and Nobody were able to maintain a humanoid form with minimal physical mutation. This is also how the members of Organization XIII appear human, despite being Nobodies, though the other members have more prominent mutations like Saix's pointed ears or Axel's facial markings.
Going further into the past, Xehanort was also able to deliberately release his heart to supplant Terra's, stealing his Body and imprisoning his Heart in the process. However, due to the powerful grudge against Xehanort, Terra's Soul was able to persist on its own, animating his cast-off Keyblade Armor as a substitute body.
In a somewhat similar vein, Sora's Heart has held several other individuals' Hearts over the years, including Ventus' and Kairi's. In the latter case, when Sora released his Heart to set Kairi's free, the process that would normally create a Heartless and Nobody experienced a hitch. Sora's Heart became a Heartless, his Body and Soul developed its own self as Roxas, and thanks to residual emotions from Kairi's Heart, his Body then duplicated and transformed to become the pseudo-Nobody Namine. These unusual circumstances are also why Namine has the ability to tamper with Sora's memories.
The precise play between light, darkness, and the components of a being are what governs the creation of Heartless and Nobodies and the balance of the realms. It could be said that neither Heartless nor Nobodies are truly supposed to exist, at least within the realm of light, but the fact that they continue to is why Keyblade Wielders like Sora and company continue their fight.