Closed headphones are designed to isolate you from external ambient noise to varying degrees. Large closed-back headphones with bigger earcups typically deliver a bit more isolation than smaller-sized closed models. Closed headphones also minimize the music leaking out of the earpieces - which will happen with open-back headphones.
Open headphones are designed to allow audio leakage out of the earpieces. They don't block out ambient noise. However, open-back models can often sound more airy, clear or spacious versus their closed counterparts. Open headphones are useful if you need to remain aware of your external surroundings.
Currently rated by our staff as the two reigning "Kings Of The Hill" in open headphones, the expensive Focal Utopia and the Audeze LCD-4 are widely considered among the world's greatest headphones ever manufactured.
Both open and closed back headphones can be either large full-size over ear models or the smaller on ear types.
Most audio professionals (DJs, mixers, recordists, studio producers) use closed headphones to allow for better concentration and isolation. Nowadays, the very best closed headphones can almost compete in sound quality with the best open headphones.