What is the connection between witches and Satan? The quick and simple answer is that there is no connection between witches and Satan. There are reasons the two are tied together, most of them are false information spread by a church looking for souls to save.
Satan is a Christian being. He was cast from Heaven because he wanted God's job instead of being a servant. He lords over other fallen angels known as demons. Simply put, he is God's opposite.
Since we know about him, let's think about what a witch is. That is a bit harder to explain because there are many different thoughts on it. If we stick with the simple definition, a witch is someone who practices magic using Nature as the core of his or her practice. It is also someone who practices Wicca, a nature based faith. Neither of which follow the devil or Satan.
Ancient witches had natural symbols that the used in their practices that were turned into false evils. The one that showed them as followers of Satan was the Horned God. The Horned God is a witch's symbol of the male, of fertility and the provider. He is often a horned stag that stands up right.
If you know the look of Satan, you can see why they get confused. Satan is a hoofed creature with horns. He is built strong and fierce. He looks a lot like the pagan witch's Horned God.
When Christianity started up the two faiths coexisted. But then the Church's Powers That Be needed more power and that meant spreading the world. The Christians figured that if they can make the nonbelievers see the evil of their ways, then they would switch over. The Christians believed that the various horned gods that were worshipped by the witches were demonic in nature. It was pointed out that Christianity's Satan was remarkably similar in appearance to the Horned God.
Of course, if the pagan witches still believed in their Horned God, they could always be killed in the name of God. Many people left the faith that had roots going back thousands of years instead of being tried as a witch. Those who did not convert went into hiding and learned quickly to practice in private.
There is, in today's world, such a thing as a Satanic Witch. Satanic witchcraft is not a religion, but is a method of magic, which is eclectic in style, created from the philosophies of Satanism, and the methods of witchcraft. Satanism is a faith that says each person is his or her own ruler. They do not worship Satan as Christians worship God. Their style and path are far different from both Christians and traditional witches.