The numbers: You'll see binoculars with numbers like 7x35 or 10x50. What do those numbers mean?
Magnification: The first number tells you the magnification strength of the binoculars. For example, with an 8x42 set of binoculars, the image you are looking at will appear 8 times larger (or closer) through the binoculars than an unaided eye.
Most birders choose binoculars with magnification of 7, 8, or 10. Higher magnification has its advantages and drawbacks. The advantage is obvious; the object you are looking at will seem bigger and closer with higher magnification.
The disadvantage to higher maginication is that it will be harder to find what you are looking for, and it will be harder to follow moving objects with higher magnification. Just in case you haven't noticed, birds tend to be moving objects.
Objective lens: The second part of the number tells you the size, in milimeters, of the objective lens. The objective lens is the large lens on the far side of the binoculars. So a 8x40 set of binoculars has a 40mm objective lens.
What size objective lens do you want? Larger objective lenses let in more light, so if you are birdwatching at dusk or in heavily shaded areas, you will want to use binoculars with a larger objective lens. The drawback to larger objective lenses is that they tend to be heavier. If you are hiking a far distance, you may want binoculars with smaller objective lenses.
For example, a 10x40 and a 10x50 will both magnify the image by 10 times, but the 10x50 will let more light in. They will also be heavier.