The Black Sun is an esoteric symbol with multiple meanings and associations. Below is a brief outline of those, but the important thing to understand is that any powerful and resonant symbol is primarily defined by how useful it can be. The best way to understand the symbol is to make it your own, and use it.
Egypt, Gnosticism, Plato & the Tao
In ancient Egypt, the god Amun (the greatest god of the national pantheon for long stretches of time) was known as a secret or invisible god, one of whose epithets was "the sun behind the sun".
Such symbolism (i.e. of a hidden sun, or God-beyond-God) came to define the centre of Gnosticism much later, although it is interesting to note that the intellectual centre of early Gnosticism and Hermeticism was also in Egypt.
Plato's writings were an extremely important influence on Gnosticism, and his principal message was of an ultimate, unknowable principle ("The Good") symbolised by the sun. Furthermore, in his Allegory of the Cave Plato drew the distinction between a first source of light that seekers after Truth might find which turns out to be merely a fire, and the truer, higher light of the Sun beyond it.
Just as Plato characterised his highest ideal as ultimately unknowable (despite it making all other things knowable), in the far east the (roughly contemporary) idea of the Tao ("path" or "way") also intimated an ultimate truth which made everything possible, but which was itself invisible or unknowable.
Alchemy and the Nigredo
As the Classical era gave way to the Middle Ages, Renaissance and Enlightenment, ancient ideas evolved into the memeplex of Alchemy. The "Great Work" of Alchemy (creation of the Philosopher's Stone, or perfection of the human condition) was divided into phases, symbolised by suns of different colours. The first sun is black, and known as the Nigredo phase. The Black Sun of Alchemy signifies death, darkness and putrefaction, but of a type which clears the way for life, light, and new growth.
The Theosophical "Central Sun"
Theosophy was an extremely popular spiritualist movement from the late 19th to early 20th Centuries. In her most popular book, "The Secret Doctrine", Helena Blavatsky (the movement's founder) wrote about a "Central Sun", an "unseen and mysterious" entity at the centre of our galaxy. Blavatsky's part in this mythological progression is important both because of Theosophy's great influence on Western esotericism in the late 19th Century, but also because it led to the development of Ariosophy, an esoteric doctrine which motivated a number of founding nazis.
1950s Nazis, Cold War paranoia & UFOs
In the immediate aftermath of World War II, a group of esoteric neo-nazis began gathering at the studio of designer Wilhelm Landig, in Vienna. They developed a mythology which had some similarities to that of the Ariosophists of 40-50 years before, and indeed to Scientology, which was being developed in the USA at the same time.
Landig explicitly referred to a "Black Sun" which in his mythology was similar to Blavatsky's central sun, a substitute swastika (the Hakenkreuz being an illegal symbol in postwar Germany and Austria), and a mystical source of energy apparently able to regenerate the Aryan race. Landig's ideas come at an interesting juncture of place and time, just as (fresh) memories of the war were giving way to Cold War paranoia and the new mythology of the UFO, both of which were natural features of Landig's novels.
1990s Neo-nazis & Subcultural occultism
In Wewelsburg castle, once the headquarters of Himmler's SS and planned "world centre", there is a room in the North Tower which was used (or at least intended for use) by the highest ranking SS officers for ritual, mystical-symbolic purposes. On the floor in the centre of that room is an inlaid marble symbol now known as the Black Sun ("Schwarze Sonne"). This symbol was not identified as the "Black Sun" before 1991, but it certainly fits the bill.
The 1990s was a time of furious memetic recombination, and these ideas exploded with the convergence of the newly-popular internet, the advent of conspiracy theory literature, and the collapse of the Soviet Union. With regard to the Black Sun meme, this recombination started with turning a handful of disparate fragments into a unified mythology, and by the end of the decade had turned into an explosion of myth-making.
Two notable 1990s subcultural developers of the Black Sun idea were Grant Morrison (comics writer who connected the meme to the Lovecraftian Cthulhu mythos) and the post-Industrial, post-Thelemite band Coil (who understood the Alchemical meaning of the symbol, and also described it as the sun seen above an imaginary world, which fits well with Plato's Idealism).
Syncretic, postmodern conspiracy theory
Some people are surprised by the "post-truth", "alternative facts" world we find ourselves living in now, but the simple fact is that they haven't been paying attention. For decades now, education systems and other societal institutions have been dismantled in the name of "liberty", and the result is an ill-educated populace who readily mistake propaganda, myth and fantasy for empirical fact.
The Black Sun meme has found itself embroiled in this postmodern fantasy-swamp for the last 10-15 years, having been adopted by any number of groups and individuals who want to attach it to their half-baked notions of Planet X, the Illuminati etc. For this reason, it is safe to say that right now 90% of mentions of the Black Sun will probably only lead to a ghetto of flat-earth nonsense of the sort we can thank David Icke for.
Black Holes, David Bowie and Blackstar
Moving forward, a couple of very interesting things have happened recently which seem likely to give the Black Sun idea enough traction to escape ignoble New Age obscurity.
One of these things is the study of Black Holes, which has undergone some remarkable developments in recent decades. Aside from some incredible implications for physics and computer science, it would seem that on some level Helena Blavatsky was right, and there is an invisible "Sun" at the centre of our galaxy.
The second thing is the recent passing of David Bowie, which really was elevated to something extraordinary and of cultural significance when the artist chose to stage his last year or two as an artwork. His final album, titled ★("Blackstar") is saturated with Black Sun imagery, in ways that are frankly too deep and complex to go into here. This really was an incredible artistic and magical act on Bowie's part, in that he managed not only to neatly sum up his life's work and attitude (particularly his well-documented but long played down occult interests), but also to simultaneously launch the Black Sun symbol into public consciousness in a way that perhaps no-one else on the planet could have done.
Philip K Dick, Asatru, & Ásentír
So now, in 2017, it seems that we find ourselves in a new era with regard to the Black Sun mythology. It has an ancient lineage to be sure, but now it also has a degree of resonance with scientific discoveries and is an extremely public symbol. The question is what this symbol will mean, going forward. As noted at the beginning of this essay, symbols find their meaning in the ways they are used, and so the question is really to what use we will put the symbol.
In early 1974, the prolific neo-Gnostic science fiction writer Philip K Dick experienced a psychotic episode of some sort which lasted several days. He spent the rest of his life trying to untangle those visions, and key parts of his notes later formed a book known as The Exegesis. A recurrent theme in that book is that of the salvific void; a liberating nothingness which stands beyond this false world of appearances (which Dick called "The Black Iron Prison").
Among the many remarkable passages in that book, one is particularly interesting with regard to the Black Sun idea. Dick was given to working through any number of weird and wonderful explanations for his experience, and at one point he conceived of a vast intelligence beyond the world as we understand it, which we are actually part of, like the appendages of some vast and strange animal "with its hand caught in a steel trap". As it turns out, the imagery of a god with a hand or foot caught in a trap of some sort occurs across multiple ancient cultures.
Because the nazis identified strongly with the ancient Germanic peoples, it is natural for people of the neopagan Asatru faith to wonder if the Black Sun symbol has some relevance to their beliefs and worship. Interestingly, it is the most ancient and powerful of the Germanic gods - Týr - who has lost his hand to the evil forces symbolised by the wolf Fenrir. It is Týr who "has his hand in a steel trap". Many people think of Odin as the most powerful and ancient Germanic god, but actually Týr is the Germanic variant of the original sky-father of the Proto-Indo-European peoples, Dyeus (known to the Greeks as Zeus, to the Romans as Jupiter, etc).
None of this is intended to argue that the Black Sun should have any particular place in Asatru, and certainly not that it had any place in the worship of the historical Indo-European peoples (although it may have). What we can say, however, is that it does have a central place in the modern, post-Asatru faith known as Ásentír, where it represents Tír, the ultimate expression of technology and expansion beyond the limits of the human condition.