Portrait photography

Portrait photography is defined as: "photography of a person or group of people, which reflects the expression, mood or personality of the subject."
Portrait photographs (much like portrait paintings) tend to focus on the face and facial features rather than including the whole body and background, though some photographers do go for this option. 

This photo of Grace Kelly (taken by Cecil Beaton) is a classic example of a portrait photo. It focuses on Grace Kelly's facial expression, which is a reflection of her mood.