Is there is a study or classification available on different types of databases? (Examples include structured, unstructured, semi structured relational, object oriented, folksonomies, etc.)
One of the key differences between the different paradigms for data management is the underlying structure used to bind pieces of data together. IMHO, the difference between structured, unstructured, semi-structured is not precise enough : the point is that the underlying structure may be more or less flexible. This formal structure is used to define the semantics of the model and so it drives everything: the query language, the concrete (in memory, on disks) data-structures, the graphical representation etc.
In the standard relational paradigm, the core concept is that of a tuple (in the set theoretic sense). XML variants are clearly tree based, RDF related models are grounded on labelled directed multi-graphs: those three models are examplified in this recent draft paper. Nowadays, one talks about stream data models grounded on infinite relations, key-values stores are grounded on functions/maps. I guess folksonomy may be multi-set oriented etc.
So "the study of different types of databases" may turn out to be "the study of formal data-structure useful for large data management".
Wikipedia has a list of database models: http://en.wikipedia.org/wiki/Database_model