Open-back headphones are for serious lovers of music. The sound is like you are in a nightclub or concert hall. Unlike traditional headphones with closed backs (trapping the sound coming from the back of the driver), open-back headphones have an open grille on the back. Thus, the sound from the backs of the headphone drivers (the little speakers inside your headphones) are able to escape out into the air. 

Because of their open backs, sound from outside can comes in, and thus open-back headphones are preferred in public places.

There are also semi-open-back headphones, with smaller grilles or vents to block more sound from the outside. Semi open-backed headphoens tend to have more bass than open-back mode.

The best open-back headphone under $500: HiFiMan HE400S (great, spacious and natural sound, versatile for great many types of sounds)

The second best open-back headphone under $500: Sennheiser HD 600 ( mellower, great for acoustic music and long listening sessions)

The second best open-back headphone for home listening: Fostex T50RP mk3 (easy-to-replace cable)

The best affordable open-back headphone under $500: Grado SR80e (a classic favorite)

The best ultra-affordable open-back headphone under $500: Fostex TH-7BB (great natural sounds at a great price)