We have the Modern Satanists, who are made of four different varieties: The Symbolic Satanists, the Naturalist Satanists, the Psychological Satanists, and the LaVeyan Satanists. Symbolic ones believe that Satan is just a symbol that has appeared throughout the ages that represents admirable human qualities; Naturalists believe that Satan is real, but instead of being a conscious being, he is simply a »dark force« in nature that manipulates all that exists. Psychological Satanists believe that Satan is the »dark side« of humanity (in a moral sense), and LaVeyans combine the three together under Anton LaVey's teachings.
Then we have the Traditional Satanists, who believe in Satan as an actual being. There are many types of them as well, the two most popular being Gnostic and Dark Gnostic Satanists. Gnostic Satanists are divided into two groups that I am aware of at this time Luciferians and Setians and they interpret Satan as the Prince of Darkness, a supernatural being who stands apart from the forces of nature and who usually represents divine knowledge and wisdom. Luciferians prefer to call him by his Roman name, Lucifer, which means »Light-Bearer« in Latin, while Setians prefer to see him as Set, the Egyptian god of Darkness.
Dark Gnostic Satanists tend to interpret Satan as the »Lord of This World,« a nature god/dess similar to the Wiccan God and Goddess, only both combined into one.
Then we have the Eclectic Satanists who combine any number of the above philosophies into their own individual brand of Satanism.