The Church of Satan was the first organization to promote the word with an associated ideology. Prior to its founding, Satanism was just a term thrown around by Christian churches to attack their opposition, which even included other churches. It had no real context before founder Anton LaVey, although Traditional Satanists will argue that they were the first.
The Church of Satan is an entity promoting an atheistic belief system that sees Satan as a powerful symbol of specific human traits, seeing humans as the real beast, and that they should embrace all of their animalistic nature. In the short life of here and now, it is our obligation to become our own gods and enjoy life to the fullest, while renouncing supernatural gods and the moral restraints that come with them. LaVey promoted that the Seven Deadly Sins are actually good things that we should embrace. He also promoted being a murderous threat to anyone who would interfere with our free will. This is called LaVeyan Satanism.
There is some confusion given that they perform spell and ceremonial rituals. LaVey didn't actually believe that these accomplished anything, but promoted them because he felt they were necessary theatrics for psychological fulfillment in place of religion. Unfortunately, LaVey himself was kind of a plagiarist (Satanic Bible) and tried to invent his own psychology (Satanic Witch). When taking this into account, the whole damn concept seems theatrical. However, there are some valid points to his philosophy surrounding natural law and survival of the fittest that many have embraced.
The CoS originally sought out to promote political advocacy through its members, however it has consistently failed to do so because of its own inaction. Instead it had only served as a source of media attention during the years LaVey was still alive, with him and affiliates being interviewed on various shows talking about their beliefs. To date, they haven't seemed to have done much of anything in terms of advocacy, and the organization has since lost members to starting their own groups.
Groups arisen from leaving the CoS over disputes and ideological differences include The Temple of Set and The First Satanic Church. The Temple of Set is covered below. The First Satanic Church was founded by Karla LaVey after her father's death leading to internal disputes in the CoS. They always have been and still are kind of a non-entity.