I'm not actually asking about the fashion items, the kind celebrities buy for their dogs... but actual outdoor designed dog boots.
What is the purpose of dog boots? Why / when would you use them?
These boots are designed for a few purposes actually.
Firstly, dogs paws can be affected by snow and ice - especially in breeds or dogs unused to colder climates. The boots help protect their paws from the colder temperatures, and also help prevent the build up of snow on their paw hair - which can then clump and freeze and cause irritation for the dog. They also help prevent damage to the foot if walking on terrain that is rather sharp. Some sled team dogs use boots, but not all.
Secondly, these boots can be used if the dog gets injured, the foot for most animals (and people, it's not always easy to keep a plaster on your foot) is hard to keep a bandage on neatly, especially when already out on a hike, and the boot can act as a way to prevent dirt and grit from getting into a cut and help the dog from over licking the area as well. In addition, in the similar way to a horse with a poultice, you can use the boot to actually keep the bandage in place.
It's not the kind of item every dog would want to wear, as many dogs don't like having something on their feet - but a friend of mine actually has a set in her dog first aid kit.