First off, read the previous sections on boots and bindings so that you know what types are available. This will enable you to choose what sort of boots/bindings you need. It basically boils down to: if you want to do carving, racing or want a very fine level of control, and aren't worried about tricks or flexibility then go for hard boots with plate bindings. If you want freeride or to do tricks in the park or pipe, then go for soft boots and bindings.
To ad-lib a quote from Mike T (newsgroup regular), the 3 most important aspects of choosing new boots are fit, fit and fit!
If the boots don't fit, you're not going to enjoy your day on the hill. Take it from me, from personal bitter experience: having boots that are too big means you can lack control of the board; having boots that are too tight means that your feet hurt like heck, and you want to go to the bar instead (and if you're doing that, you might as well have stayed at home to get drunk). It took me 3 pairs of boots before I finally found the right pair - I would save you that pain and money.
Ideally go and try the boots on in the store after a day of riding,
when your feet are swollen from their normal size (feet naturally do
this, so you want to accomodate for it). Get boots that are just too
tight at that time of day. Firstly the boot will (should) pack out a
little, and your feet will be a little smaller at the start of the
day... but please, change your socks before going to the
store. Phewwww!