Short answer
In Islam, Satan is Iblis, a jinn, not a fallen angel. Jinn are beings Allah (﷾) made from fire, before He made man. When Allah (﷾) told the angels to bow to Adam, Iblis refused out of pride. He said he was better, being made of fire, while Adam was made of clay. Allah (﷾) gave him time until the Day of Judgement, and he swore to lead people astray. But he is not a rival god. He has no power to force anyone, only to whisper and call. On the Last Day he will tell his followers to blame themselves. Over those who believe and trust in Allah (﷾) he has no authority. Muslims seek refuge in Allah (﷾) from him, and Allah (﷾) is enough as a protector.
Full explanation
Iblis was a jinn, made from fire
The Qur'an says that when the angels were told to bow down to Adam, they all bowed except Iblis. It says plainly that he was one of the jinn, and that he broke his Lord's command. The jinn are a creation Allah (﷾) made before mankind, from the fire of a scorching wind, a fire free of smoke. Angels, the Qur'an says, never disobey Allah's (﷾) commands, so Iblis could not have been one. The word shaytan (satan) means a rebel. The Qur'an uses it for Iblis, and for any evil one among jinn or men.
“Behold! We said to the angels, "Bow down to Adam": They bowed down except Iblis. He was one of the Jinns, and he broke the Command of his Lord. Will ye then take him and his progeny as protectors rather than Me? And they are enemies to you! Evil would be the exchange for the wrong-doers!”
“We created man from sounding clay, from mud moulded into shape; And the Jinn race, We had created before, from the fire of a scorching wind.”
“And He created Jinns from fire free of smoke:”
“O ye who believe! save yourselves and your families from a Fire whose fuel is Men and Stones, over which are (appointed) angels stern (and) severe, who flinch not (from executing) the Commands they receive from Allah, but do (precisely) what they are commanded.”
“Likewise did We make for every Messenger an enemy,- evil ones among men and jinns, inspiring each other with flowery discourses by way of deception. If thy Lord had so planned, they would not have done it: so leave them and their inventions alone.”
His sin was pride
Allah (﷾) asked Iblis what kept him from bowing when he was commanded. He answered, "I am better than he: Thou didst create me from fire, and him from clay." His first sin was pride, and a judgement of worth by what someone is made of. Adam and Eve also disobeyed when he tempted them, but they repented and were forgiven. Iblis did not repent. He asked for time until the Day of Judgement and swore to lie in wait for people on the straight path, from every side.
“It is We Who created you and gave you shape; then We bade the angels prostrate to Adam, and they prostrate; not so Iblis; 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."”
“He said: "Give me respite till the day they are raised up." (Allah) said: "Be thou among those who have respite." He said: "Because thou hast thrown me out of the way, lo! I will lie in wait for them on thy straight way: "Then will I assault them from before them and behind them, from their right and their left: Nor wilt thou find, in most of them, gratitude (for thy mercies)."”
“And behold, We said to the angels: "Bow down to Adam" and they bowed down. Not so Iblis: he refused and was haughty: He was of those who reject Faith.”
“They said: "Our Lord! We have wronged our own souls: If thou forgive us not and bestow not upon us Thy Mercy, we shall certainly be lost."”
“Then learnt Adam from his Lord words of inspiration, and his Lord Turned towards him; for He is Oft-Returning, Most Merciful.”
He can call, but he cannot force
Satan is not a second god fighting Allah (﷾) as an equal. He lives only by the time Allah (﷾) has given him. The Qur'an says he has no authority over those who believe and trust in their Lord. His hold is only on those who take him as a patron. On the Day of Judgement he will say to his followers: "I had no authority over you except to call you but ye listened to me: then reproach not me, but reproach your own souls." So no one can say that the devil made him do it. Each person answers for his own choices.
“(Iblis) said: "O my Lord! because Thou hast put me in the wrong, I will make (wrong) fair-seeming to them on the earth, and I will put them all in the wrong,- "Except Thy servants among them, sincere and purified (by Thy Grace)." (Allah) said: "This (way of My sincere servants) is indeed a way that leads straight to Me. "For over My servants no authority shalt thou have, except such as put themselves in the wrong and follow thee."”
“No authority has he over those who believe and put their trust in their Lord. His authority is over those only, who take him as patron and who join partners with Allah.”
“And Satan will say when the matter is decided: "It was Allah Who gave you a promise of Truth: I too promised, but I failed in my promise to you. I had no authority over you except to call you but ye listened to me: then reproach not me, but reproach your own souls. I cannot listen to your cries, nor can ye listen to mine. I reject your former act in associating me with Allah. For wrong-doers there must be a grievous penalty."”
How a Muslim guards against him
The Qur'an calls Satan an open enemy, to be treated as one. He watches people from where they cannot see him, as he once tricked Adam and Eve. Muslims are told to seek Allah's (﷾) protection from him before reading the Qur'an. And the last chapter of the Qur'an is a prayer for refuge from the whisperer who withdraws. When doubts come, such as "Who created your Lord?", the Prophet (ﷺ) told Muslims to seek refuge in Allah (﷾) and stop there. He also said every person has a companion from the jinn. Allah (﷾) had helped him against his own, he said, so it urged him only to good.
“Verily Satan is an enemy to you: so treat him as an enemy. He only invites his adherents, that they may become Companions of the Blazing Fire.”
“O ye Children of Adam! Let not Satan seduce you, in the same manner as He got your parents out of the Garden, stripping them of their raiment, to expose their shame: for he and his tribe watch you from a position where ye cannot see them: We made the evil ones friends (only) to those without faith.”
“When thou dost read the Qur'an, seek Allah's protection from Satan the rejected one.”
“Say: I seek refuge with the Lord and Cherisher of Mankind, The King (or Ruler) of Mankind, The god (or judge) of Mankind,- From the mischief of the Whisperer (of Evil), who withdraws (after his whisper),- (The same) who whispers into the hearts of Mankind,- Among Jinns and among men.”
Abu Hurayra reports that the Prophet (ﷺ) said that Satan comes to a person and asks who created this thing and that, until he asks who created his Lord. When that thought comes, the person should seek refuge in Allah and stop there.
Abdullah ibn Mas'ud reports that the Prophet (ﷺ) said every one of them has a companion from among the jinn assigned to him. They asked whether he did too. He said yes, but Allah helped him against it, so he is safe from it, and it commands him only to good.
The Christian picture of Satan
Many Christians teach that Satan was once a great angel, Lucifer. He rebelled against God (﷾) out of pride, they say, and was cast down from heaven with the angels who followed him. They read this in Isaiah's words, "How art thou fallen from heaven, O Lucifer, son of the morning!" They also read it in Revelation's dragon, cast out with his angels, and in Jesus' words, "I beheld Satan as lightning fall from heaven."
“How art thou fallen from heaven, O Lucifer, son of the morning! how art thou cut down to the ground, which didst weaken the nations!”
“And the great dragon was cast out, that old serpent, called the Devil, and Satan, which deceiveth the whole world: he was cast out into the earth, and his angels were cast out with him.”
“And he said unto them, I beheld Satan as lightning fall from heaven.”
Isaiah was speaking of a king
Isaiah's chapter says who it is about. The prophet is told to "take up this proverb against the king of Babylon". The "son of the morning" is that king, who boasted he would exalt his throne above the stars and was brought down to the pit. "Lucifer" is the Latin word for the morning star, the name the poem gives the proud king. The verse is not about the origin of the devil at all. The Qur'an needs no such reading. It says in plain words who Iblis was and how he fell.
“That thou shalt take up this proverb against the king of Babylon, and say, How hath the oppressor ceased! the golden city ceased!”
“How art thou fallen from heaven, O Lucifer, son of the morning! how art thou cut down to the ground, which didst weaken the nations!”
“For thou hast said in thine heart, I will ascend into heaven, I will exalt my throne above the stars of God: I will sit also upon the mount of the congregation, in the sides of the north: I will ascend above the heights of the clouds; I will be like the most High. Yet thou shalt be brought down to hell, to the sides of the pit.”
“Behold! We said to the angels, "Bow down to Adam": They bowed down except Iblis. He was one of the Jinns, and he broke the Command of his Lord. Will ye then take him and his progeny as protectors rather than Me? And they are enemies to you! Evil would be the exchange for the wrong-doers!”
Perspectives
Christian
Many Christians teach that Satan is Lucifer, a fallen angel. He rebelled in heaven, they say, and was cast out with other angels.
Isaiah 14:12 describes Lucifer, son of the morning, fallen from heaven.
Rebuttal
Isaiah says the passage is a proverb "against the king of Babylon". The fallen "son of the morning" is that king, brought down for his pride. The verse says nothing about where the devil came from.
Revelation says the devil was cast out of heaven with his angels.
Rebuttal
Revelation is a book of visions, full of dragons and beasts, and Christians themselves read it in many ways. The Qur'an says in plain words that Iblis was one of the jinn, made from fire. And it says the angels never disobey Allah (﷾).
The Islamic perspective
Satan is Iblis, a jinn made from fire, who refused out of pride to bow to Adam. He was given time until the Day of Judgement. He can only whisper and call, and he has no authority over those who believe and trust in Allah (﷾).
The Qur'an says Iblis was one of the jinn, and that the jinn were made from fire before mankind.
Iblis refused to bow to Adam because he thought himself better, being made of fire.
On the Day of Judgement Satan will say he had no authority over people except to call them. They should blame themselves, he will say.
Satan has no authority over those who believe and trust in their Lord.
Muslims seek refuge in Allah (﷾) from Satan, before reading the Qur'an and whenever doubts come.