What is the function of the tail section on a fixed-wing aircraft?
First off lets be clear that a tail section does the same thing on small aircraft that it does on large aircraft. 

Second off lets be clear that a tail section is not necessary for flight and there are planes like the Northrop Grumman B2 that dont have a tail section they are affectionally called "Flying Wings".

The short answer is that the rudder (vertical part) controls yaw and the elevator (horizontal part) controls pitch. A plane is capable of movement and similarly adjustment in all 3 dimensional axes. But lets dig in a bit:

The Elevator: This is usually located in the back but by no means has to be. Burt Rutan has implemented many designs that have forward mounted elevators. As to functionality they essentially allow you to alter the pitch of the plane which in turn alters the angle of attack of the wing which furthermore has a direct effect on the lift generated by the wing. In a simple world we could say that when you pitch up you climb and when you pitch down you descend. The reality is that the engines thrust plays a large factor in this. For example if you are flying at a level attitude and increase thrust your will increase speed (and the speed of air over the wing) which will generate greater lift and cause the plane to elevate with out a pitch increase (you will have to apply some force to keep the plane level though). The elevator provides a way to easily alter the pitch of the plane (and angle of attack) so you can climb when you want. Generally speaking you will also add power for the climb. For a more lengthy explanation check out the wikipeidea article.

The Rudder: The rudder is mainly to control the left-right yaw of the aircraft. In a single engine prop plane this can be used to help overcome the left turning tendencies created by uneven force of a spinning prop (P-Factor). In a more general case the rudder is often used to correct yaw when a heavy crosswind is present on landing. It is also used in flight to ensure your turns neither skid or slip (stay coordinated) remember "Step On The Ball". For more reading check out Wikipedia   

This picture shows the axes of movement pretty well.