Can your instructors point you to reference works so you can prepare for the examination? Were any handouts provided? Do you have access to a good design library?

There are so many different systems to categorize fonts. Take a look at Luc Devroye's extensive overview at
http://cg.scs.carleton.ca/~luc/classify.html

In Switzerland a lot of schools still teach the DIN system or Bosshard's modified version of it. Three books that are often recommended:
Schriften erkennen: Sauthoff, Wendt, Willberg (Verlag H. Schmidt)
Technische Grundlagen zur Satzherstellung: Hans-Rudolf Bosshard (Niggli)
Buchstaben kommen selten allein: Indra Kupferschmid (Niggli)

Indra Kupferschmid's book contrasts the historical classification (DIN) and the classification according to formal principles (static/dynamic) which Gerrit Noordzij described in detail.

http://www.designhistory.org/type_milestones.html
http://tkb.design.fh-aachen.de/03_01_timeline.php (in German unfortunately).
http://blackboard.tynecoll.ac.uk/nln/frameset.php?url=Arts%20and%20Craft...
http://www.rsub.com/typographic/timeline/

For extended info on the characteristics you could look into the descriptions that come with the Vox classification and work from there.