In a way binoculars are like two small telescopes. The same rules apply for binoculars as do for telescopes. There is in most cases one big difference. Since most binoculars do not allow you to change their eyepieces, advertising the power, or magnification factor is appropriate. Since aperture is also an important factor, manufacturers often advertise binoculars with two numbers, magnification and aperture. For example you may see binoculars advertised as 10x50 or 16x80. The first number is the magnification factor and the second number is the aperture in millimeters.
Another thing to know about is the quality of the lenses, and the coatings used to improve performance. There are other features that some more expensive binoculars have such as image stabilization, and interchangeable eyepieces.
Just as with a telescope, when using some larger binoculars a mount is important. If the binoculars are high power it is nearly impossible to hold them steady enough to view comfortably. Small movements will cause huge movements in the image.