Essentially, headphones come in two fundamental designs. They have either open-back or closed-back designs, features that emerge as extremely important when it comes to investing in headphones for your auditory experience.
Comparing these two designs is comparing two major functions and hence, two different types of people. The closed-back design, which is the most common build of headphones, essentially serves one important purpose i.e. noise isolation. These headphones have ear-cups that are enclosed with metal and/or strong premium grade plastic. These are enough to seal in the music you are listening to and cut out any ambient noise. For noise cancellation even beyond this, there are noise cancelling closed-back headphones that are a step more than noise isolation.
With a closed-back headphone, you will not hear the noise that surrounds you, making it appropriate for noisy environments like buses, trains and traffic in general. As there is no sound leakage, these headphones allow you to wear them out and about without disturbing fellow travellers or pedestrians. If you enjoy uninterrupted and unadulterated intimacy with your music, then closed-back headphones are the best for you.
However, the major drawback of closed-back headphones is that they will never be sonically as good as open-back headphones. The reason for this is that as the headphone drivers move front and back, there are sound waves directed at the internal parts of the cups, thus causing undesirable reflections that hinder with the sound. The other drawback that makes these headphones inappropriate to be used while running/cycling/walking in a public place is the fact that you are so isolated from your surroundings, that you may not hear important signals, such as a car approaching while you are taking a walk on the road.
Open-back headphones are the exact opposite of their closed-back contemporaries. This design allows air to pass through your headphones, making them breathe and giving you an experience like listening to live music, where you are conscious of the sounds and noises around you. Since the open-backed design reduces any reflections from within the housing as well as improves the airflow for the drivers' free movement, it makes the music sound a lot more natural. It improves the timbre considerably and also widens the soundstage.
What emerges to be the drawback of such a design is that there is a good amount of sound leakage, making these headphones inappropriate to be worn in public, as it can be disturbing to the people around you. But the same feature makes it a great design as on the road, you will always be aware of what is going on. Another drawback is the lack of isolation letting all kinds of background noise in.
Open-back headphones are headphones that can't really be used outdoors because they leak sound. They're best for use for leisure home listening or in a studio for monitoring. You can't record with open-back headphones though.
These two fundamentals of a headphone are tremendously important as when you are investing in a pair, you decide how you want your music to come to you. Both designs are great in their own way; hence the choice lies with you as an individual and your unique and personal music inclinations.