Correct spelling, explanation: this is the correct spelling because it is a combination of fare (good fortune) and well. The word in Middle English was spelled farwel. Due to its pronunciation [fair-wel], it may be mistaken as fairwell but it’s an incorrect form. Farewell is the only correct spelling.

Definition of farewell:
noun, someone’s departure; wishes of good fortune
For his farewell, we saw him off at the airport.
When Smith retired we organized a surprise-farewell party for him.

Incorrect spelling, explanation: fairwell is incorrect. It is probably a mistake made because of the pronunciation of farewell (the correct form), which is [fair-wel]. However farewell comes from the word fare and well, so it is spelled this way.