Raine's recipe will indeed make a "fizzy" ginger beer (or ginger ale, if you want). Very much a soda pop type beverage.
It will have some alcohol in it, but only a trace amount from the yeast, which are only there to provide carbonation. That's also why you have to store them cold, otherwise they'll keep eating sugars and you will continuie to get a "drier" tasting beverage until the bottle can no longer contain the pressure. Then you get a bomb. Seriously.
Another note - don't store them in the fridge indefinitely. The cold only severly slows the yeasties, it doesn't stop them completely.
As for the bottles, you can re-use old crown cap beer bottles if you have a capper, or swing top bottles (Think Grolsch beer). You can also re-use plastic PET bottles, but they can be hard to get the odor of the previous soda out of there (Especially a Cola or ROOT BEER).