Binocular Power
Learn what it is!
Power (or magnification) is extremely important but don't just look for that and forget the rest. More power will mean larger binoculars, which means more weight which means you may tire very quickly of carrying a large pair let alone try to held them for any length of time.
The power rating increases with the number - a 7x is not as powerful as a 10x and a 20x is more powerful than both.
The best power for bird watching is found in the 7x to 10x range. This power range will give you enough magnification to do the job of identifying and observing birds without the problem of just being too heavy to hold steady. Shaky hands makes for difficult bird watching. And a sore neck from carrying a large set will soon dwindle all your enthusiasm for that hike in the woods to nothing.
The three most common powers people buy are 7x35, 8x42, and 10x40.
The second number after the x refers to the diameter of the objective lens [the front lens] of the binoculars. This numbered measurement is given in millimeters The larger the number given, the more light is allowed in. And lots of light available means you get a bright, clear image to look at through the eyepiece.
Of course, someone then runs out and buys a 12x50 set of binoculars - all the power AND lots of light, right? Yes BUT you had better buy a tripod as well before you leave the store. The size of lenses and prisms required will make for one very heavy pair of binoculars. You might even want to consider a little red wagon to haul it all around as well!

Note that the higher the power, the lower will be your field of vision as well. Very few experienced birders go beyond a 12x power for their binoculars as they become too difficult to find the bird and focus quickly. And since they also become heavier at the higher power the more likely you hand will shake which will make the image jumpy to your eye.
-10x [or 12x] will work best for distant shore birds or high flying gulls but you may need a tripod or even switch to a spotting scope for best results.
-7x works well for close-up work
-7x or 8x works well in low-light or night conditions for subjects such as owls, feeder birds or woodcocks
-7x works well when watching a number of high flying hawks but you can get better detail on one hawks with a 10x - so what you want will determine your binocular power
-7x with a larger field of vision will help you find a bird much faster than a 10x