Due to the individual grounded nature of satanism, it is not useful to try to find the one definition of it. I cannot say anything about other people's satanism, except those who told me. Therefore, everything you read here, is how I understand it. I understand that this is a rather mindfucking view about it. To bring it in context, there's also an article about my views on LaVey Satanism.
About the Word
שטן, Satan; Adversary, Accuser, Disbeliever …
The word Satan is hebrew and can be translated to “Disbeliever” or “Adversary.” And this is mainly how I understand it intellectually, in a very consequent way. I prefer to accept the fact, that wisdom is impossible, as everything is perceived and therefore not known a priori.
About Knowledge
In the, ironically named, Ages of Enlightenment, mankind started to ignore that fact. Science is taken as truth. And not as a paradigma, as it should. Though the scientific knowledge changed over years, got falsificated in it's sense of reality and proved in it's sense of reality (In short words: It evolved, like every other craft) many people seem to be not aware that science doesn't tell truth, but probabilities. Everything is seen as true, unless proved otherwise. Does something become true because there's no evidence against it? I deny that.
The Path of the Left Hand
“Better to reign in Hell than serve in Heaven.”
John Milton's Paradise Lost
Satanism is meaningless, if it is not lived. The path of the left hand is, therefore it's name, something to be walked, not stared at and talked about. It is about individual power, in any sense. I'd go even further and claim it is about being God. About stopping to seek a meaning of a life of someone else. Someone created by the environment, depending on the environment.
Satanism is about Creation, in every phase of life, asleep or awake, in enragement and solitude. And what is Creation? Beginning and End, Life and Death, Alpha and Omega. You have to Destroy to Create, you have to Die to be Reborn.
To sound much more occult, Satanism is about living your true will. As I understand it, true will is a force in material and non-material spheres, and satanism is to live the material side.
Satanism and Occultism
For many people occultism and satanism do not fit together. The problem is, that the occult arts are often recognized as a search for the divine. But that's not how I would describe it.
It is right, though, that the occult literature is full of paradigms, centered about serving some divine. But those are not the only ways to practice the black arts, they are more those that are searched by the people.
Occultism in it's abstract sense means Power (at least to me), and there are much more ways in it, than a single mind can hold. Satanism is about understanding and therefore becoming God. This "sort of behaviour" is mostly associated with the material planes, but I see no sense in serving an atheistic dogma that is mostly based on ignorance of - as it seems - impossibilities.
Simply said, satanism is about reigning over oneself, but as I understand it, not about only the physical incarnation, as about the whole existence itself.