Class also decides what weapons a character is proficient with; an untrained character cannot even aim (Mass Effect) or use (Mass Effect 2) a weapon. It also determines what armors they may wear, and what other talents they possess.
In Mass Effect, specialized class talents are unlocked once you have completed the assignmentUNC: Rogue VI on Luna. There are six specializations, two of which are available based on your initial class choice. Selecting a specialization offers access to a new trait that add ranks 7-12 to your class specific talent; ranks 1-6 are unchanged, and grant the same bonus when you take ranks 7-12.