Greetings!
Karl gives good advice above... if you are planning on buying only 1 pair of binoculars, a 7x42 or 8x42 would probably be your best choice because of it's better low-light performance than 8x32 models (at the cost of higher weight however - nothing's free!).
If you are buying 2 or 3 pair for different purposes, I would recommend that you buy these, in this order:
1 - a 7x42 or 8x42 for general purpose use
2 - a 10x25 compact pair from Nikon, Swarovski, or Zeiss for hiking or glovebox use.
3 - a 10x50 for low-light use
Those 3 will serve you well for every conceivable situation, with option #1 (7x42 or 8x42) being the most useful all-around. If you become a "binocu-holic" like many people on this forum, there are many other configurations for specialty uses - including image stabilized, ultra-high power (12x+), built in laser rangefinders, and switchable magnifications (Leica Duovid). While most of these are pretty neat technically, their relatively high cost and specialized functions make them dubious choices as a first birding binocular to buy.
One final thought... many people prefer high-power binoculars because of the style of birding they do... if you are watching a lot of raptors, waterbirds (from shore, not from boats) or birding in wide-open spaces, consider 10x42 as your main choice instead of 7x or 8x. If you spend much time in forests, valleys, or birding on boats, however, the 7x or 8x are definitely superior.
For world travel, if I ever have the opportunity to go "world birding", I would take along 2 binoculars - a pair of 7x42 and 10x25. This would combine maximum optical flexibility with minimum weight.
Best wishes,
Bawko