The first number you see on a pair of binoculars represents the magnification. If a pair of binoculars is 10x25, then 10 is the magnification, meaning that the object will appear 10 times bigger than it does to the human eye.
Some binoculars have two first numbers, such as 12-36x70 or 25/40x100. When the first number is listed with a hyphen (-) it means that the binoculars have the ability to zoom in magnification. In the case of the 12-36x70, this means that the binoculars can change from a magnification of 12 all the way up to a magnification of 36. If the numbers are separated by a "/" this does not mean the binoculars are zoom but rather that they use more than one fixed-power eyepiece. In the case of 25/40x100, the binoculars have one set of eyepieces that can produce a magnification of 25 and another set of eyepieces that can produce a magnification of 40.
The second number refers to the objective lens diameter in millimeters. In the 10x25 example, the objective lens is 25 millimeters, or about 1 inch in diameter.
The objective lens is the lens that is pointed toward the subject. It gathers light to make the subject brighter. The size of the objective lens determines the size of the binoculars and the larger the objective lenses, the more light the binoculars will gather. As the size of the objective lens grows, however, the heavier and more expensive a pair of binoculars will be. For birding and wildlife viewing, an objective lens of at least 40 millimeters is recommended. Full size binoculars with an objective lens of at least 40 will perform better in situations where the light is less than ideal and will produce steadier images and a wider field of view. Small or midsize binoculars with an objective lens size of 25 to 35 will weigh less and be best for backpacking.
The exit pupil is the size of the hole in which light will travel through the eyepiece and into the eye. One can identify the exit pupil by looking for the bright circle in the center of each eyepiece. The exit pupil is calculated by dividing the objective lens number by the magnification number. In general, the bigger the exit pupil, the brighter the image. If you plan to use your binoculars during low-light conditions like dawn and dusk or if you will be using them to view stars and planets in the sky, you will benefit from binoculars with a large exit pupil.