Black Sun Sonnenrad Symbol
The Black Sun, also known as the sonnenrad (sun wheel) in German, specifically comes from the floor of the north tower of Wewelsburg Castle, which was renovated by SS-leader Heinrich Himmler. The castle was a meeting place of only highest members of the SS, and Himmler considered it the axis mundi (center of the world) for their ideology.
The specific meaning of the symbol, if it even had one to Himmler, is unknown. There aren't even records of a name associated with this symbol. There is no suggestion that he called it the Black Sun; that term was associated with it afterward.
Origins
Himmler was highly interested in Germanic folklore and pagan belief and therefore appears to have adopted the symbol from similar historical shapes. While his Black Sun specifically has twelve arms, historical versions varied widely in the number of radiating spokes.
The historical versions are considered by many to be sun wheels, similar to sun crosses, which is why calling this symbol a sun comes easily to most. (However, one should note that the Wewelsburg symbol is made of green stone, not black.) The center of the Wewelsburg symbol also originally had a gold center to it, which is a common solar symbol.
Sun symbols commonly represent triumph, life and goodness, and sun symbols with central points to them often represent unity and centrality as well. All of these meanings fit well within Nazi ideology and worldview: the unity of a single race centered around a powerful party and leader triumphing over lesser, oppressive, evil races, embracing the fulfillment of life and goodness as defined by the Nazis.
Meaning of the Radiating Spokes
There are a variety of possible meanings in the design of the spokes. Germanic sun wheels commonly do have bent spokes. For Himmler, the bent nature was likely important because each spoke represented the Germanic sowilo rune of the Elder Futhark, which represented the sun. Himmler adopted a modern rune system which called the symbol sig and had it represent victory. His most known use of the sig rune is the insignia of the SS, which uses a double sig rune.
The pattern created by the crooked radiating spokes might also be interpreted as three overlaid swastikas. This interpretation has led some neo-Nazis to adopt the symbol, particularly in places where the display of swastikas are illegal.