This term from the theatre world describes someone speaking with an expressionless face. It is often used to describe a type of humour, in which the joke is delivered in a flat impassive way, which contrasts with the words or humour.
Photo source: Thomas Ruff who has produced many artistic photos imitating the passport photo
Shelley was brilliant, the way she led you to believe she was the wife of a sheikh, all with that deadpan expression of hers.
He proposed to her in such a deadpan voice that she thought he was talking about an upcoming business meeting, not their dream life together.
Pan is a slang word for face, hence the combination with dead to describe this look which gives nothing away. It is one of the forms of humour that people find most difficult to pick, understandably.