'Leaving headroom' means leaving a space between the top of the subject's head and the top edge of the frame, usually when taking shots head-on. It is related to the Rule of Thirds; the idea is that your subject's eyes should lie on the upper third-line of the frame. Like the Rule of Thirds, it's roots lie in painting.
The concept is related to 'looking room': if you have a subject looking to the right, you should leave some space on the right hand side of the frame for the subject to 'look into'.
As ever, these are rules of thumb, not absolute laws.