What is Vee on a 74hc4051 mux?

The 74HC4051 has level shifters inside that allow you to use a logic supply that is GND to Vcc and another supply voltage Vee that is less than (or equal to) GND.

This is a great advantage when you want to use dual supplies such as +/-5V for the analog electronics, and the digital signals are 0 to +5 (or +/-3.3V with digital logic 0/3.3V), and if you don't need the feature you can just ground Vee.

I am not sure I believe the operating area shown here:

The early CD4051 clearly required Vee <= GND.

The CD4051 mux can work at +15 volts single-ended supply, or +/- 7.5 volts for bipolar supply handling bipolar signals to +/- 5 volts or so. The IC also needs a ground ref for the 3 to 5 volt logic used to select which of 8 input/outputs is selected to connect to the common I/O pin.

So for bipolar operation you need 3 power supply voltages and a gnd connection. The Vee pin is the negative power for the analog section. For bipolar operation it is connected to -7.5 volts. For single supply operation Vee is connected to ground. Even powered by +/- 7.5 volts for the CD4051, it is best to keep an analog signal in the range of +/- 5 volts, at the channel resistance goes way up close to the supply rails.

The 74HC4051 is limited to 11 volts single ended supply or +/- 5.5 volts for a bipolar supply. The signals are limited to about +/- 3.5 volts.