I found this diagram in the TI TLV431 datasheet. The diode symbol looks like a zener, but has an addition input on the side. What is this supposed to represent?
This symbol could represent any number of precision adjustable shunt regulators. Combined with a precision op-amp they are often a voltage reference for other circuits.
Typical internal reference is 1.24 volts, so by connecting Vref between 2 resistors, one to cathode and one to anode, it behaves like an op-amp feedback to multiple the Vref times the resistor ratio.
Connecting Vref to the cathode as the drawing shows gives it no gain, so its clamp (shunt) voltage would be 1.24 volts.
These type of adjustable zener diodes are often used with the opto-couplers LED input to set the output voltage of SMPS type power supplies, as part of a feedback loop. This allows for galvanic isolation between the high-voltage primary and a low voltage output(s).
The title of the datasheet you linked is, "TLVH431, TLVH432 Low-Voltage Adjustable Precision Shunt Regulators". It's a kind of programmable Zener diode.
The description tells us
The TLVH431 and TLVH432 devices are low-voltage 3-terminal adjustable voltage references, with specified thermal stability over applicable industrial and commercial temperature ranges. Output voltage can be set to any value between VREF (1.24 V) and 18 V with two external resistors (see Figure 19). These devices operate from a lower voltage (1.24 V) than the widely used TL431 and TL1431 shuntregulator references.
When used with an optocoupler, the TLVH431 and TLVH432 devices are ideal voltage references in isolated feedback circuits for 3-V to 3.3-V switching-mode power supplies. They have a typical output impedance of 0.25 Ω. Active output circuitry provides a very sharp turn-on characteristic, making the TLVH431 and TLVH432 devices excellent replacements for low-voltage Zener diodes in many applications, including on-board regulation and adjustable power supplies. [Emphasis mine.]
From the datasheet.
The flyback power-supply is one application where it is useful. We need to signal from the secondary side back to the swithing controller on the primary side when the output voltage reaches the required voltage. Opto isolation is good and the TLVH makes it easy. The control pin is referenced from a voltage divider on the output and the resistor values are calculated to give 1.24 V on the control pin when the output voltage is correct. The device then turns on allowing current to flow through the opto-LED.
The TLV431 is a versatile reference device, which can be regarded as a programmable zener diode.
If you are happy with the very over-simplified description of what silicon junction transistors do, that is for Vbe < 0.7v they are off, and for Vbe > 0.7v they turn on, then a TLV431 is very much like that.
For Vra (voltage from reference to anode) < 1.24v they are off, and for Vra > 1.24v they are on. In contrast to a transistor, where on and off are not well defined, and the 0.7v is strongly temperature sensitive, in a TLV431, the 1.24v is very stable with temperature, and the current gain is very high.
With a resistive divider between anode and cathode to the reference, the overall effect is to give a very well defined (low impedance) voltage drop, when shunting a current.
As it happens, you can use a voltage divider with a junction transistor in exactly the same way. However the resulting shunt regulator is neither as low impedance or as temperature stable as a 431. This sort of shunt regulator is often found in class B audio amplifiers precisely to make the bias to the output devices track their temperature.
The zener symbol actually represents an integrated circuit which behaves like a very accurate and stable zener diode because it contains all the elements (transistors, resistors, diodes, capacitors) necessary to keep the voltage drop across it constant over a wide range of temperature and currents through it.
The third pin, usually called "Reference" or "Program" pin, can be tied to the cathode pin and the IC can then be placed into a circuit like a standard zener diode od 1.24V, 2.5V or whatever the minimum voltage is for the particular IC). Using a voltage divider with 2 resistors or a trimmer/potentiometer, you can set any voltage across the "diode" part of the IC, from the minimum to the maximum supported by the IC.
There are integrated circuits which only have 2 electrodes - anode and cathode, just like a zener diode, but actually contain many components inside which make them much more stable and accurate than regular zener diodes. With those, you are limited to their voltage set during manufacturing, without later adjustments possible, and they can range from 1.24V to 33V.