Humor and humour are the same word: humor is the American spelling, humour the British one, and both cover being funny, a person's mood, and going along with someone to keep them happy. Think of the 'u' as the British badge, the same pattern as colour and flavour.
The core difference
Humor is the American spelling and humour the British spelling of an identical word, covering both funniness and mood.
- humor — US spelling, the quality that makes something funny: The film's humor is dry and easy to miss.
- humour — UK spelling, a person's mood or state of mind: He was in a foul humour after the meeting.
- humour (verb) — to go along with someone's wishes to keep them happy: She humoured her grandfather by listening to the same story again.
How to tell them apart
Both spellings descend from Latin humor, meaning moisture or fluid. Medieval medicine held that four bodily fluids, or 'humours', governed a person's temperament — which is why the word can mean both 'mood' and, later, 'funniness': a good or bad humour was originally about which fluid was dominant in you.
The spelling choice is purely regional: American English drops the 'u' (humor, like color and flavor), while British and most Commonwealth English keeps it (humour, colour, flavour). Match your spelling to your audience and stay consistent — don't switch between humor and humour within the same piece.