Magnification, or "power" is the first number listed on a binocular. It indicates how many times larger an object appears through the binocular versus the naked eye. A 8x binocular magnifies your view eight times. In general, an increase in magnification causes a decrease in brightness.
Objective lens is the second number and tells us the size (in millimeters) of the lens at the end of the binocular is. In general, the larger the number, the more light enters the binocular, providing a brighter image.
Close focus is the shortest distance that you'll be able to see in focus. If you are interested in birding in woods or your backyard, or viewing butterflies, you'll want a binocular with a short close focus.
Field of view is the widest area you can see when you look through the binocular. If you plan to use the binoculars to scan fields or over water, consider a binocular with a large field of view.
Eye Relief is the distance (in millimeters) from eyepiece lens to where all the light rays come into focus. Only eyeglasses wearers need concern themselves with it. On average, an eyeglasses wearer needs at least 15mm of eye relief.