Headphones are small loudspeakers which are worn very close to, or even in, the ear. 
Headphones are useful for a number of purposes: 

If you are recording a sound and want to monitor the signal going to or from tape, using a loudspeaker is problematic as it's sound will be picked up by the microphone and the potential exists for feedback or signal distortion. Use of headphones will provide the acoustic isolation you need and minimize the influence of room acoustics. 

Headphones come in several designs with closed, open, and in-the-ear "bud" types commonly in use. 
The closed design is constructed so as to isolate the listener somewhat from the surrounding acoustic environment. These are useful when you want to hear more accurately what the signal sounds like with minimal interference from the ambient acoustic surroundings.  The open design minimizes acoustic isolation so as to allow a lighter weight and more comfort, as well as allowing the user to hear the surrounding soundscape more clearly mixed with the electroacoustic signal.  In-the-ear headphones provide the lightest and perhaps most inconspicuous type of headphone use with minimal acoustic isolation. These are useful for onstage monitoring by performers. 
Drawbacks of headphone use include ear fatigue and a difference of stereo perspective from loudspeaker listening. When listening to a stereo program with headphones the differences between left and right will be accentuated.  Caution should be exercised when using headphones. Extended listening at high levels can result in ear fatigue, threshold shift, and even long-term hearing loss.