Under what conditions should we use negative voltage?
It depends on the application. I'll take two examples, the first is that of the wired telephone lines from the local exchange to the Customer Premises equipment and the second being the use of an Amps for AC amplification.

But before that, I will assume that you are comfortable with the entire concept of Negative Voltage. Simply put, it's the reference point (zero) of the system being taken on the positive terminal of the battery.

Telephone Lines: The main reason lies in electro-chemical theory. Usually copper wires are used in Telephony and the lines are buried in ground. The minerals in the ground around the telephone wire can be more reactive than copper (see  Metal Activity Series  ), and in the presence of moisture, the electrolytic reaction corrosion of Copper (acting as Anode) into ground (impurities as cathode) when Copper is at higher potential as compared to the anodes (Ground). 
Now, if the Copper wire is held at a lower potential (relative to ground), this reaction will not happen. Hence, this justifies our first application of Negative Voltage. For more info on this, see this.

Now, we come to our more native (to electronics domain) application of Negative Voltage. A  Push Pull Amplifier. see here for simple schematic. Now, in a push pull amplifier, usually only one of the two transistors is ON at any given time. The one connected to the positive cycle pushes the positive cycle of the AC (amplifies the positive cycle). Now, with the polarity of AC reversed in the next part of the cycle, this amplifier will not work because it is not biased for that operation. These transistors are biased to have their DC operating points near zero (because of symmetric power supplies of opposite signs). Thus during the negative cycle, only the Transistor with Negative Power supply biasing comes into play and Pulls the voltage down (thus amplifying the negative cycle).
This is done to amplify large AC signals because if we were to bias a single transistor at some point in the middle of its DC Load line curve, then the range of Voltage swing that would be amplified faithfully by the transistor would be severely limited.

Similarly, in many Opamps, we need Positive and Negative Voltages for faithful amplification and operation. Negative voltages would be needed in almost all of those applications where the input voltage will have significant excursions below the negative range and we desire amplification. This is not usually the case with Audio Amps as the audio signal (base-band)usually stays above the 0 DC level. 

Thus, the application of Negative Voltage is usually application specific, and the other times to properly bias the active elements of the circuit.

Hope it helped.