A brief definition of the basic specification for any binocular, e.g. 8 x 40, or 8 by 40, is in order. The first number is the magnification power of the binocular. It will make an object appear that many times larger than the naked eye. So, an 8 power binocular magnifies an image 8 times. The second number is the diameter of the larger, or "objective" lens in millimeters. This is the lens that gathers in the image of the object you're looking at. So, an 8 by 40 binocular has a 40 millimeter objective lens. The larger this lens size, the larger the binocular body. This can be a factor if you can't carry around something very heavy for several hours outside.
Magnification power can often be a subject of debate among birdwatchers. Most hand-held binoculars are in the 7 to 12 power magnification range. 7 and 8 power are considered a general-use range because they usually have a larger field of view, which is the maximum angle a particular model of binocular can see. This is usually expressed as a number of feet in width that can be seen at a standard distance from the binocular (usually 1000 yards). A good range for field of view is 350 to 400 feet.
Higher magnification binoculars (9 to 12) have relatively lower field of views - lower 300 feet range. But, if you're mainly using them for birdwatching, this is not often a big problem. So, if you want to use binoculars for a wide range of uses (nature viewing, sports events, scenery, etc.), a lower magnification is fine.
If you mainly want to use them for birdwatching, a higher magnification binocular will enable you to see more detail. Besides field of view, another trade off of using higher magnification binoculars is that the image will often shake a bit more, especially if you have unsteady hands.
The quality of the glass, construction, and lens coating system makes the biggest difference in the price of the binocular over any other factor(s). Better quality binoculars will present much sharper, brighter images to the viewer, which makes it much easier to identify whatever wildlife you're looking at. You want binoculars with fully, multi-coated lenses and BaK-4 glass where possible. Some of the best features are ED (extra low dispersion) and HD (high density) glass. Basically, you should buy the best binoculars your budget can afford. In the long run, you will be far happier with your decision.
So, the key is to try a lot of different models to find the binocular that feels right in your hands and is easy to look through for your specific vision. This way, you can find a great pair that you will be very happy with for years.