The first part of the numbers listed on our binoculars refers to the magnification level, the second part refers to the diameter of the objective lens. The larger the magnification level the larger an image will appear to be, the larger the objective lens, the brighter and sharper the the result image will appear
If we were to compare a pair of 7x50 binoculars with a pair of 8x21 binoculars the 7x50 will make an object appear as if you were 7 times closer to it. The 50mm objective lens will produce a brighter clearer image, the 8 x 21 will produce a make the object appear larger but it would not be as clear or as sharp as the 7x50's
The larger the objective lens the larger the binoculars are, so you could slip a pair of 8x21 binoculars in a bag or coat pocket with no problem, however a pair or 7x50 would be much larger and would probably need to be carried in a bag, or over the shoulder or hung around the neck.