In 8 x 42, ..
8 (Magnification):

This number indicates how much closer the binoculars bring distant objects. For example, a magnification of 8x makes something look eight times larger than with the naked eye.

42 (Objective Lens Diameter):

This is the diameter of the larger, front-facing lenses in millimeters. A larger objective lens, like the 42mm one, gathers more light, resulting in a brighter image, especially in low-light conditions

Similarly in 10 x 50

Usually we say that 8 x 42 is optimum. Reason being more the magnification, steadier our hands need to be. The subject we see in the binocular may shake if magnification is too much and our hands are shaky. If magnification is very high as in astronomical scopes etc, one may have to mount the binocular or scope on a steady tripod or other mount.

50 Diameter may gather more light than 42 and appear brighter, but overall binocular will be heavier and costlier (for the same quality).

So we need to strike a balance between magnification, light gathering, weight, cost etc. So usually we say 8 x 42 is optimum balance.

But you can always think about the above parameters and decide what you want and choose accordingly.