When looking at binoculars you will see the figures 7x50 or 10x21, etc. The first number refers to the magnification...seven times or ten times, for example. The higher the magnification power, the more detail you see but the steadier your hand needs to be. If you are jacked in closer than 10x you will see wonderful details but with a much narrower view and a decline in depth of field...if you can hold your hand steady enough to catch any of it.
The second number is the diameter of the front lenses (aka "objectives") in millimeters. The idea is the larger the number the brighter the image. If you are going to do some heavy duty dusk and night time viewing, you will want a 50 mm objective lens. But keep in mind there is a tradeoff...the larger the lens the larger the binoculars and their price tag.
Take the objective lens diameter and divide it by the magnification and you have the "exit pupil" figure. For example, 21 divided by 10 gives you a 2.1 mm exit pupil.
Binoculars are designed either with a roof-prism or a porro-prism. The light travels in a straight line through objective lens and eyepiece in a roof-prism pair. With a porro-prism setup, the objective lens and eyepiece are offset but you get a little brighter view. Roof-prism binoculars are more compact and tend to cost more than the porro-prism binoculars.