Terminology - What do the numbers mean?
All binoculars will have two numbers to designate them, such as 10x50. This means a binocular with 10 times magnification and 50mm objective lenses (the big ones!). Common sizes are 8x42, 10x50, 20x60, 15x70, etc.
Optical glass is generally BAK4 or BAK7 standard, BAK4 is denser and gives superior vision.
Optical coatings are applied to the lenses (and prisms in good binoculars) to improve contrast and other image properties, coatings are what make the difference between a cheap and a good binocular.
The cheapest coating is Ruby, these are red and generally horrible. A single magnesium flouride coating is a blue colour and a step up from ruby. The best optics have green multi-coatings, sometimes just on the main lenses but often on all air-glass boundaries inside the binocular too.
The size of the beam of light coming out the eyepiece is calculated by the lens size / magnification, for a 10x50 set this will be 50/10 = 5mm. Your iris opens to 6 or 7mm depending on your age. A set with a small exit pupil is a bit like an 8" TV, you can see the picture but can't make out any detail. Binoculars with lens sizes below 30mm are useless in this regard.
The amount of light a binocular gathers is based on the lens size squared (from the area of a circle, PI x R^2). So a 50mm set gather double the light of a 30mm set, 70mm double that of a 50 and so on. For astronomy you need at least 50mm, preferably 70+.
A zoom binocular has a lever to alter the magnification and might be called 10-30x60, it's 60mm and goes from 10-30x magnification. You can't hand hold anything above 20-30x without it shaking too much. But many larger binoculars allow you to attached an adaptor for a camera tripod. As you increase the zoom the field of view will decrease, you'll see a smaller area.
Zoom binoculars offer high power but are generally not as good optically as fixed magnification sets. Anything offering a power over 35 is hopeless in the real world unless it's tripod monted and has big lenses.
What should I buy for my application?
General use / Birdwatching / hill walking: 8x32, 8x42, 10x42.
Astronomy: 7x50, 10x50, 15x70, 20x90, any 100mm.
Sporting events, 10x50, 12x50, 10x42, 8x32
High power observation, 20x50, 20x60, 20x90, 100mm, 60 or 70mm zoom.
What should I avoid?
Avoid anything with a lens below 30mm, you'll get a dim image with no detail.
Avoid ruby lenses and BAK7 glass, the image quality is poor.
Avoid overly powerul binoculars with a small exit pupil. For high power you need big lenses, 60mm+.
How much should I spend?
At least 30.00 for a basic set. Beyond that don't be fooled by price, it is NOT a guide to quality in binoculars. I know of sets costing 30 and 250 with the exact same spec. from the same factory. Look for good (green) optical coatings, BAK4 glass and a sensible specification.