Most Christians, and most people who mention Satan, think of him as none other than Lucifer. They regard him as an Angel of Light and Beauty, and one who fell from Heaven primarily because of Pride.
The first reference of Lucifer comes from the Old Testament in Isaiah 14: 12-17:
How you are fallen from heaven, O Day Star, son of Dawn! How you are cut down to the ground, you who laid the nations low! You said in your heart, ‘I will ascend to heaven; above the stars of God I will set my throne on high; I will sit on the mount of assembly in the far north; I will ascend above the heights of the clouds, I will make myself like the Most High.’ But you are brought down to Sheol, to the depths of the Pit. Those who see you will stare at you, and ponder over you: ‘Is this the man who made the earth tremble, who shook kingdoms, who made the world like a desert and overthrew its cities, who did not let his prisoners go home?’
This text is actually in reference to an ancient Babylonian King, by the name of Nabuchadnezzar, who exiled the Jews from Jerusalem. Christians say that no mortal man could ascend his throne above the heights of God, so they attribute this quote to Lucifer.
Another quote from the Old Testament, found in Ezekiel 28: 12-19:
"Son of man, take up a lament concerning the king of Tyre and say to him: "This is what the Sovereign Lord says:
"'You were the model of perfection, full of wisdom and perfect in beauty. You were in Eden, the garden of God; every precious stone adorned you: ruby, topaz and emerald, chrysolite, onyx and jasper, sapphire, turquoise and beryl. Your settings and mountings were made of gold; on the day you were created they were prepared. You were anointed as a guardian cherub, for so I ordained you. You were on the holy mount of God; you walked among the fiery stones. You were blameless in your ways from the day you were created till wickedness was found in you. Through your widespread trade you were filled with violence, and you sinned. So I drove you in disgrace from the mount of God, and I expelled you, O guardian cherub, from among the fiery stones. Your heart became proud on account of your beauty, and you corrupted your wisdom because of your splendor; I made a spectacle of you before kings. By your many sinds and dishonest trade you have desecrated your sanctuaries. So I made a fire come out from you, and it consumed you, and I reduced you to ashes on the ground in the sight of all who were watching. All the nations who knew you are appalled at you; you have come to a horrible end and will be no more.'"
The connection with the King of Tyre and Lucifer is also Christian in origin.
Satan is mentioned in the New Testament by the Christ figure, Jesus, as one who fell from heaven. But again, there is no direct reference to Satan being Lucifer:
"He replied, "' I saw Satan fall like lightning from heaven."
-Luke 10:18
There are other stories relating to Satan/Lucifer from various cultures and belief systems. The Qur'an describes him:
"It is We Who created you and gave you shape; then We bade the angels prostrate to Adam, and they prostrate; not so Iblis (Lucifer); He refused to be of those who prostrate."
(Allah) said: "What prevented thee from prostrating when I commanded thee?" He said: "I am better than he: Thou didst create me from fire, and him from clay."
-Qur'an 7:11-12
According to Islam, Lucifer fell because not only of pride, but his refusal to bow to men, who were made of clay, him of fire.
Michael W. Ford, occult author and Magus of the Order of Phosphorus, clearly points on the Adversary's history in various cultures and beliefs such as Zoroastrianism and Manichiaean lore.
"Zoroastrianism and Zurvanite religions along with the later Manichiaean doctrines describe a sorcerous spirit and god called 'Ahriman' or 'Angra Mainyush', he who tore from his mother's womb before his brother, Ahura Mazda, or the God of Light."
-Adversarial Light, Magick of the Nephilim, Michael W. Ford, page 33
Michael goes on to explain that Set is a rebel figure in Egyptian mythology.
Many things, figures, and entities throughout history can be attributed to Satan, the Adversary, or even Lucifer, depending on personal perspective. But there is no factual evidence that Satan is indeed Lucifer.
Luciferians see Lucifer as a symbol of divine power and inspiration, who represents our intellect or higher being. Lucifer also brings light to achieve balance with the darker aspects of human nature, and exists within the mind of all true Satanists and Luciferians.