What “Bun” means
Bun is the Patois form of "burn", in the literal sense. In relationship slang, to "bun" someone means to cheat on them, and to "get bun" means to be cheated on. Bun is also the spiced loaf eaten with cheese at Easter, a separate word from the English "bun".
Examples
Nuh eat di whole bun
Don't eat the whole bun
Shi bun im.
She cheated on him.
Mi get bun.
I got cheated on.
How to use it
The cheating sense is casual slang, common in songs and gossip; it can be a sensitive accusation, so use it with care. The food sense is neutral. "Bun" is also used in Rasta and dancehall speech to reject or condemn something (see bun fire).
Where usage varies
Context separates the senses: talk of Easter means food, talk of a relationship means cheating, and "bun" plus a target ("bun dat") is a rejection.