I pulled this from the FAQ over at /r/satanism, written by /u/modern_quill, as they could explain it better than I ever could. All credit goes to them:

This is a simple question, but it has a complex answer because it depends on who you ask. Satanism as a philosophy and religion was first codified by Anton Szandor LaVey in his 1969 publication of The Satanic Bible. Some people refer to this secular Satanism as "LaVeyan Satanism" as a nod to Anton LaVey. The Satanic Bible borrows from the works of Might is Right by Ragnar Redbeard, Ayn Rand's Objectivism, and Frederich Nietzsche's Der Wille zur Macht. This is the most widely practiced form of Satanism and is championed by the Church of Satan (CoS) to this day. At its most basic definition, "LaVeyan Satanism" is about living the best life that you want to live, and bending the world around you to your will to achieve that goal. A Satanist sees themselves as their own God. There is, of course, much more to Satanism than that very basic definition, but we expect people to do their own research as well. Most LaVeyan Satanists will simply call it Satanism, as there is only one form of Satanism from the Church of Satan's perspective. Members of the recently formed secular organization called The Satanic Temple (TST), by comparison, see Satanism as political activism. The Satanic Temple often makes news headlines with their efforts to establish a separation of church and state and do not include The Satanic Bible as part of their organization's canon, but rather The Revolt of the Angels by Anatole France. Later, some people in the United Kingdom split from The Satanic Temple to form the Global Order of Satan (GOS). There are also theistic Satanists, some believe in a literal Satan and some do not. There are also more esoteric organizations such as the Temple of Set (ToS), which was formed by former Church of Satan member Michael Aquino after infighting within the organization in 1975 caused many theistic members to split away and become Setians.