What determines the voltage at which a capasitor will discharge at. Is it the supply voltage at which it is charged at?

 

The voltage on a capacitor must be continuous, so whatever the voltage was immediately before it started discharging is the voltage it will start discharging from.

While this will be the voltage it was charged to, it may or may not be the value of the supply voltage used to charge it. For instance, if you have an RC circuit with a time constant of 10 seconds and you start the discharge after 10 seconds, then the capacitor will not be charged all the way to the supply voltage, but rather only about 63% of the way.

 

A capacitor will start to discharge at any voltage if there is a circuit for it to discharge through, if the voltage on the capacitor is higher that the voltage on the circuit. Likewise, the capacitor will charge if the external voltage is higher than the capacitor voltage.
The charge or discharge will stop when the voltages are the same. So, for example, you have a charged capacitor and then place a discharged capacitor across it, the first will discharge into the second until the voltages are equal.
A resistor across the capacitor will cause it to discharge until the voltage is zero.
All capacitors have an equivalent internal resistor "across" it so, over time, they will self discharge.

 

What determines the voltage at which a capasitor will discharge at. Is it the supply voltage at which it is charged at?

What determines the tea level in a cup at which you can take a drink? Is it tea level in a teapot?