The most important step in choosing a pair of binoculars is to figure out the power you want. Binocular power is represented by two numbers, as in these examples: 6x30, 7x35, 8x26, 10x50, and 12x36. The two numbers are not directly related; the first expresses the magnification and the second expresses the diameter of the objective lens.
-- Magnification --
The first number, which is always the smaller of the two, corresponds to the amount of magnification that the binoculars allow for. This measures how much nearer the observed image will appear when looked at with the binoculars. Even though bigger sounds superior, that's not always the case with binoculars. Picking the correct magnification will vary depending on your intended use.
Lower magnification binoculars, 7x and lower, provide for a wider field of view which can make targeting objects at a distance—like birds, animals, or ships—easier. A bigger field of view also makes it a lot easier to follow motion of the object being observed. Since they take in more of the scene, they bring in more light. Because of this, the resulting image will be easier to see and brighter even though it is not as greatly magnified. Because of the lower magnification, they are also less sensitive to vibration or movement. If you think you are going to be in the wild using your binoculars without a tripod, this can be an important consideration. Lastly, lower magnification binoculars are typically able to focus on objects that are closer by than equivalent higher magnification binoculars.
But of course, lower magnification binoculars ultimately provide less magnification, and consequently reduced detail than higher magnification models. If you really need high magnification binoculars, typically 11x and higher, for your use, make sure that you additionally purchase a tripod or get binoculars with image stabilization. The higher the magnification, the more affected the binoculars are to vibration. Once you get up to 10x magnification binoculars, they will be nearly unusable without an image stabilization system or a tripod. Larger power binoculars also tend to be heavy. So if you intend to use them for a long length of time, a tripod will be a more comfortable choice than binoculars with image stabilization.
As for pricing, it shouldn't be too surprising that higher magnification binoculars tend to be more more costly than equivalent lower magnification models. Unless you have special needs or uses intended for your binoculars, a moderate middle-of-the-road magnification amount is 8x. Binoculars with this level of magnification result in a reasonable trade-off of magnification while still providing reasonable cost, good brightness, and low weight.
-- Objective Lens Size --
The second number in the power corresponds to the diameter of the objective lens in millimeters (mm). The objective lens is the lens in the front of the binoculars, away from your eye. Larger objective lenses tend to indicate binoculars that are larger overall in size.
Binoculars with objective lens sizes of 30 mm and under are usually found on pocket and compact models. While pocket and compact binoculars tend to be lighter and easier to carry, the downside to reduced lens size is that the smaller lenses allow less light into the binoculars. This can result in darker images that are harder to see. If you intend to use your binoculars at night, binoculars with smaller objective lenses are probably not a good choice.
If brightness, or nighttime use, is an important consideration, objective lenses that are 50 mm or larger may be your best option. However, these binoculars will almost always be heavy and large. If you plan to use binoculars of this size, you will want to have a tripod available, independent of the magnification level.
The most common objective lens sizes used tend to be between 35 and 50 mm. Since the magnification level also has an effect on the brightness of the image (higher magnification is darker), the objective lens sizes tend to increase as magnification increases to balance out the lost brightness. For a good general rule, shop for binoculars that have an objective lens diameter that is at least 5 times bigger than the level of magnification. For example, 6x30, 9x45, 12x60, etc. While some binocular manufacturers have exceedingly good optics that will result in good brightness at a ratio under 5, the safest thing is to stay at or above this ratio for adequate brightness.