An accident is something that happens unexpectedly or by chance, especially something unpleasant, undesirable.

1. Her father was killed in a car accident.
2. There was a serious railway accident near London yesterday.
3. He had an accident at work. A crate fell on him and injured his shoulder. (Such accidents are officially called industrial accidents.)
– John’s left the door unlocked.
– I’m sure it was an accident.
– I’m afraid I’ve broken a glass.
– Oh, don’t worry. Accidents will happen.

By accident is used in the same sense as accidentally in such sentences as:

6. Our luggage was sent on to Rome by accident.

An incident is an event, especially one of relatively minor importance. It is not necessarily unexpected or unpleasant.

7. There were several amusing incidents during the journey. In one of them Alan got off the train to buy a newspaper and nearly got left behind.
8. She told us about an incident in her childhood which had made a deep impression on her.
9. He tends to exaggerate the importance of minor incidents.

Incident is also used more specifically to denote a relatively minor hostile act, for example, a protest, an attack, a clash between small numbers of troops.

10. The Conservative candidate was shouted down in an incident at last night’s election meeting.
11. A bomb exploded in a department store in Londonderry yesterday. No one was killed but several people were injured in the incident.
12. There have been several border incidents during recent weeks.

With reference to a literary work (a story, novel or play) incident has practically the same meaning as episode, that is, a piece of action which can be considered separately.

13. The incident in the restaurant shows Brian’s character very clearly.

Incidental means “occurring as an occasional part, accompanying but not forming an essential part.”

14. That conversation was purely incidental. Everything had been decided long before.
15. He was given an extra £50 to cover incidental expenses.

Incidentally means “by the way”.

16. I’m sure you’ll enjoy the book. Incidentally, the author went to the same school as my brother.