Highlight headroom is the measure of how much additional capacity a given photographic media (such as film or digital) has in order to record the detail within the brightest parts of a scene.
As an example, consider a photograph of a white wedding dress against a white background. With limited highlight headroom, it will be hard to appreciate the intricate details within the fabric of the dress. The higher the available headroom, the more subtle shades of white will be captured.
In practice, photographers are often faced with photographing outdoors on a cloudy day, but with the sun shining through a gap onto the subjects forehead. This can often lead to a bright reflection which records as bright white. If this bright spot (called a highlight) lies at the edge of the persons outline (from the cameras perspective) then it may end up blending into the cloudy sky if there is insufficient highlight headroom. This is a particular problem with digital cameras which only have a limited gamut of discrete colour values.