Open air are designed so that the rear of the diaphragm is open to the environment. They offer no noise cancelling and you can hear things in the room around you.
Closed are the traditional headphones that you are most likely familiar with. The diaphragms are in housings and the housings and are typically sealed to the ear with leather or plastic pads.
Once you determine which design is best for you, then start looking at specific brands and models.
www.head-fi.org is a useful site, although I try to avoid it.
Headphone.com is also useful. They have graphs of frequency response vs SPL for most of the major brands and models.
