What is the difference between jinn, demons and shaytan?

Jinn are a creation of Allah, shaytan is a role, and "demon" is only an English word that does not match either one exactly. Jinn are beings made from smokeless fire who, like humans, can believe or disbelieve. A shaytan is any rebellious, evil being, whether jinn or human. Iblis is the chief of them.

Jinn: a creation with free will

The Quran says that Allah created the jinn before humans, from smokeless fire:

وَٱلْجَآنَّ خَلَقْنَـٰهُ مِن قَبْلُ مِن نَّارِ ٱلسَّمُومِ "As for the jinn, We created them earlier from smokeless fire." Quran 15:27

Jinn are unseen to us, have free will, and are accountable like humans. Some are Muslim and righteous, and some are disbelievers. So "jinn" is not a word for evil beings; it names a whole kind of creation. See What are jinn in Islam? and Can jinn be Muslim?

Shaytan: a rebellious devil

The Arabic word shaytan describes a devil, any being that is rebellious and evil and leads others to evil. It is not a species. The Quran says:

وَكَذَٰلِكَ جَعَلْنَا لِكُلِّ نَبِىٍّ عَدُوًّا شَيَـٰطِينَ ٱلْإِنسِ وَٱلْجِنِّ "And so We have made for every prophet enemies—devilish humans and jinn." Quran 6:112

So there are human shayatin as well as jinn shayatin. When Muslims say "ash-Shaytan" (with "the"), they usually mean the chief devil, Iblis.

Iblis: where he fits

Iblis refused to prostrate to Adam when Allah commanded it. The Quran says he was one of the jinn, who rebelled against his Lord's command (Quran 18:50). He was not an angel; angels never disobey Allah. His descendants and followers are the shayatin among the jinn. See Is Iblis an angel or a jinn?

What about "demons"?

"Demon" comes from Christian and older European traditions, where it often means a fallen angel or an evil spirit. Because Islam sees jinn as a creation with a choice, and shaytan as a role, the English word is only a rough match. Most people use "demon" to mean evil jinn or shayatin, and that is fine as long as you do not mean fallen angels or all jinn.

A quick summary

Jinn: a creation made of fire; some good, some evil.
Shayatin (devils): evil, rebellious beings, among jinn and humans.
Iblis / ash-Shaytan: the chief devil who refused to prostrate to Adam.
Angels: a separate creation made of light, who never disobey.

The way to be protected from the whispers of shaytan is described in What dua protects me from shaytan?

Key points

Jinn are a creation of Allah made from smokeless fire, with free will.
Shaytan means a rebellious devil, from jinn or humans.
Iblis was one of the jinn, not an angel.
"Demon" is only a rough English word.

Tip: Do not call every bad thing the work of jinn. Many problems are ordinary human ones.