The second number (50 in 7X50) is the diameter of the binocular's objective lens in millimeters. The objective is the lens that you point at the object you're trying to see. Bigger objectives gather more light, so a 7X50 binocular shows you more in dim light than a 7X35.
The exit pupil is the ratio of lens diameter and magnification. In case of 7x35 it is 5 mm, in case of 7x50 it is more than 7 mm. The dark adapted eye of a child can be as large as 8mm, adult have decreasing pupil diameter and at an age of 50 normally not more than 5mm. The result would be that for children, especially newborn, the 7x50 or 7x60 is optimum. With increasing age one should choice a 7x35 or 10x50, if larger magnification is necessary. The choice of binocular is, however, also driven by a variety of other factors (next section).
The second number has a *big* effect on size, weight & cost; *30 are much smaller, lighter, cheaper than *50, but in dim light (dusk & dawn) noticeavly brighter; in good light you wont notice the difference. I've switched to 8*40 from 8*50 for birdwatching.
There is *absolutely no point* in having the ratio of the two numbers (ie 8*30 about 4mm, 7*50 about 7mm) greater than ~7mm, because that is the size of the pupil of the eye wide open; you do see daft adverts for binocs with silly ratios. Two other important factors - get ones with good coatings, multiple reflections can mess up the image, especially looking towards (NOT AT!) the sun; 'eye relief' is *very* important if you wear specs. Other than that you pretty much get what you pay for in optical quality!
Most kids would have no hope of holding a 7*60; just too big & heavy, they are far better off with a dimmer or smaller image which is actually in the field of view & not shaking all over the place! Very few people can comfortably hand hold mag. >=10, & unless experienced have trouble locating the subject too; many people of 'increasing age' would find 10*50 a pain to use. Even for an experienced adult, 8*50 becomes a pain when tired, cold, its windy etc let alone if you are on a boat! In such cases 8*30, 7*35, 8*40 will be much better choice depending on the balance of mag & brightness you need; on a boat etc I'd go for 7* for sure. In most cases the bigger aperture brings remarkably little benefit except for short periods at dusk & dawn, under heavy overcast etc because the pupil is stopping down. You pay the money, cart the weight about, & suffer the image shake the rest of the time! But of course they make you look like a super twitcher etc; you just dont actually see as much.....
Of course, if you are going to put it on a tripod its a different matter, but *finding* & tracking things in the narrow field of view of high mag. can still be an issue.
Most of the big binoculars (>10X) have significantly larger objectives on them, such as 15X70, 15X80, and 20X80. That extra volume of glass, and the metal to hold it in place, makes for one heavy pair of binoculars.