The Headsman is an Imperial executioner who wears a set of unlootable and unplayable executioner's robes, an executioner's hood, boots, and gloves, and carries the headsman's axe. He is one of the first NPCs you'll meet in gameplay, during the initial quest, Unbound.
After you create your character at the start of the game and join Ulfric Stormcloak, Ralof, and the other prisoners near the chopping block, you'll watch the headsman decapitate one prisoner before you're called up for your turn. You are forced to make your way to the chopping block and lean down to see the headsman lift his axe to execute you, but are saved just in time by a dragon attack. The headsman is then knocked down by the force of Alduin landing on the tower nearby. His fate is unknown but it seems safe to assume that he is killed in the attack.