What are the functions of these components in this 15V 500mA power supply?
Bz1 is a self-oscillating warning buzzer, and the circuit around R1, T1 &amp; Bz1 is an over-current warning.
The current from the power supply drawn by the load you connect to it ends up returning through R1. When the current increases sufficiently to cause enough of a voltage-drop across R1, T1 begins to turn on which in turn causes Bz1 to buzz at you.  

You are correct that D1 is a 15V 1W zener. It is the reference voltage source for the power supply circuit.  

T4 and T2 together provide negative feedback to control the output voltage of the power supply, based on the voltage from P1 and D1.
P1 is simply acting as an adjustable voltage-divider to allow you to adjust the voltage from 0V to 15V.
If the voltage at the output exceeds the voltage at T4's base exceeds around 0.6V, T4 will start to conduct. This in turn causes current to flow through R6 - raising T2's base voltage - causing it so to conduct.
T2 pulls the voltage at T3's base down, causing it to start to turn off (or conduct less), resulting in the output voltage being lowered.