What are the ways of classifying bread?
The Bread Baker's Apprentice has a 2-page long chart of different types of breads; in terms of classifying (not "describing") them, they look at the following:


Water Content - literally the amount of water in the bread. From lowest to highest you have Stiff, Standard, or Rustic.
Richness - the amount of fat, primarily, but also sugar, milk, eggs, or anything else to provide a richer mouth-feel.
Fermentation - direct or indirect specifies whether or not there is pre-fermentation.
Leavening - various categories describe the type of leavener used, including commercial yeast, wild yeast, chemical leavening, or a mix.
Flatbreads have their own designation and can also describe unleavened breads.


That's enough to classify the vast majority of breads out there.

As far as "describing" bread goes, that is entirely subjective and culture-dependent. For anything more detailed, you really need to go and read a book - there's far too much literature to condense into a single Q&amp;A thread.