Boots are sized in Mondo sizing. Here is a sizing chart for your convenience. In general, beginners want a boot that directly corresponds to their street shoe size. Intermediates should go a half size smaller. Experts should go a full size smaller.
The higher the number, the stiffer the boot. Skiers lighter in weight and beginner to intermediate skiers tend to prefer a boot with a softer flex. Conversely, heavier and more aggressive skiers are more likely to prefer a stiffer boot.
Measured in millimeters, the volume refers to the last, or width, of the boot. Narrower boots are best for expert, aggressive skiers, who care more about performance than comfort. Foot width factors in as well, with narrow-footed skiers prefering boots with narrower lasts. Keep in mind that in general, the narrower the boot the less comfortable it will be.
A demo ski is a used ski that comes with a mounted binding that can be adjusted to any boot size. Demo skis typically come from demo fleets or programs, whether on the mountain, in a mountain town shop, or from the manufacturer.
A new ski is brand new and has never been on snow. New skis can come either with or without bindings. If a ski is referred to as "flat" it means the ski does not come with bindings.
Alpine touring boots are far better for backcountry skiing than regular ski boots. AT boots feature walk mode and light weights, plus soles made for walking.