1) What do you need?
For most people, you use a binocular for general purposes: spectator sports, sight seeing, etc. Except for some special activities, the most important criteria you need in choosing a binocular is portability and ease-of-use. If a salesperson tries to convince you otherwise, walk away.
The first specification we need to decide is magnification as indicated above. For all around purposes, 5x to 10x pretty much covers everything. It is a common mistake to assume that the higher the power, the more "powerful" your binocular is. The fact of the matter is you will be exhausted in a few seconds looking at the "dancing" image through a 15x binocular, unless you want to carry a tripod everywhere you go. Some opera glasses offer 2x and 3x power, but you definitely will not regret it if you take a regular 7x one with you instead. Another advice: never buy a zoom binocular! You may think it is useful to change the power of magnification, but it's cumbersome, difficult to use, and easily broken.
The second number is the Aperture of the objective lens. In a 7x50 binocular, the objective lens diameter is 50 mm. It is true that the larger the objective lens you have, the more light you gather from the object you are viewing. However, a larger objective lens also means a longer and heavy binocular body. So, you need to balance portability and lens size. I have boxes of binocular samples from manufacturers, but I find myself usually put a 7x50 or a 12x42 compact binocular in my pocket before leaving home.
There is also a practical limit on the size of the objective lens. This is determined by the exit pupil of a binocular. For a 7x50 binocular, divide 50 mm (objective lens size) by 7 (power of magnification) gives 5 mm, which is the size of the exit pupil. If you want to take full advantage of a large objective lens, i.e., to collect all the light funneled through the binocular, the binocular's exit pupil cannot be larger than your eye's entrance pupil (you know, the black hole at the center of your eyeball). This is where the limit comes in. The entrance pupil of a human eye changes with the light condition, but the maximum size is limited by age. The entrance pupil can reach 8 mm for a teenage, but is only ~4 mm for a 50 year old. For an entrance pupil of 5 mm, a 7x40 is just as good as a 7x50, but the latter is much heavier.
Of course, there are special needs. For example, experienced bird watchers prefer a 40 or 42, even 50mm objective lens to gather more light and see more details in deep shadow. A mariner likes our 7x50 because it is more forgiving when the deck is constantly moving.