Boots, are probably the most important piece of equipment you will buy because you want to avoid slips, trips and blisters. The key things to look for when buying a pair of boots is that they are waterproof and have a bellows tongue, a good sole, and not to many seams.
Waterproof boots can either be made of leather and have waterproofing applied to them such as Nikwax, which must be reapplied from time to time. Or have a lining of breathable water proof material such as Gore-tex under the leather or nylon. This allows sweat out and keeps water from coming in. Plastic boots are also available for alpine / winter mountaineering, whilst these can be worn for summer mountain walking in Britain I find mine to warm for this use.
A bellows tongue also helps to keep feet dry. The tongue is connected to the boot along all the tongues length, the idea being that if you step into water the only way it can get into the boot is by running down your ankle into the boot. If the boot had a normal tongue and you stepped into water it would get in as soon as it reached the base of the laces.
Soles should have a deep tread and a square heel. Avoid boots that have a sloping edge at the heel as they will cause slips when going down hill.
Seams should be kept to a minimum because the stitches will eventually let water in if you don't keep them waterproofed and can cause rubbing.
Height of the boots, this is a matter of taste really and depends whether you want your ankles covered or not. If they aren't then its easier for the ankle to bend when crossing steep terrain, thereby making it easier to walk. On the other hand because the ankles aren't supported you are at slightly more at risk of straining the ankle. Padded cuffs around the ankle part of the boot do make the boots more comfortable and so are worth looking for in a pair of boots.
Buying boots, you should try them on before you buy. During the day your feet swell, especially if its hot and you walk along way. So the best time to try boots on is in the afternoon after you have walked a mile or two. Take with you the socks you will wear and try both boots on and walk around and ideally up some stairs. Also, you should be able to comfortably wiggle your toes.
Shanks, some boots have shanks in the sole. These are basically a strip of steel in the sole of the boot that helps keep the sole rigid and protects the sole of the foot from rocks as you walk over them. The shank can be 1/2, 3/4 or full. The full prevents the sole of the boot from bending at all the aim being to make climbing using the toe of the boot easier and also to make crampons fit properly. Whereas, the 1/2 and 3/4 shanks allow the boots to bend at the toes which makes walking up hill easier than the full length shank which is like walking up hill with ski boots on. Which means you have to adjust you walking technique. If your boots haven't got shanks then don't worry to much as they are only important if your going to be walking over rocky terrain for several days. If you think the soles of your feet may be sore after walking over rocky ground then you can buy insoles, but remember your boots will be a tighter fit then, so try them with the boots before you buy. The advantage of insoles in the winter is that they insulate your feet from the cold.