Traditionally, op-amps were bi-polar devices, early op-amps operating with ±100 V DC outputs provided by vacuum tube electronics. Later solid-state op-amps settled on ±15 V DC power supplies to produce ±10 V DC outputs. As "head room" improved, power supplies were reduced to ±12 V DC for the same ±10 V DC output. Since the original purpose of op-amps was to solve differential equations, both positive and negative output signals were mandatory. Today there are many circuits where bi-polar outputs are perhaps convenient but not necessary. As
@Gryd3 stated, it depends entirely on the purpose.
If you are contemplating using a circuit that produces, or accepts, both positive and negative signals with respect to some common or "ground" potential, then ±DC power supplies should be your first choice, typically ±15 V DC. However, many modern op-amps work at reduced voltages, and if you don't need a full ±10 or ±12 V DC "swing" in the outputs, a lower voltage, dual-polarity, power supply such as ±5 V DC or even ±2 V DC might be appropriate for you.
When can you "get by" with using a single power supply to power an op-amp? Just about any time you wish, although it could make signal processing more difficult. The op-amp itself doesn't know squat about where its power comes from: it has a positive power supply terminal and negative power supply terminal. Most op-amps don't have a "common" or "ground" terminal, so as long as the voltage difference between the +V and -V supply terminals is within the specified operating range of the particular device you are using, it doesn't make any difference
to the op-amp where that voltage comes from. It can and will make a lot of difference in how you connect external circuitry to the inverting, non-inverting, and output terminals of the op-amp, but that's another problem.
If you do need a bi-polar signal and you have only a single power supply, you can create a "virtual ground" anywhere between the extremes of the two power supply terminals. That means if you have, say, a 24 V DC power supply you can create a "virtual ground" half-way between the output terminals so the positive terminal is +12 V DC with respect to the "virtual ground" and the negative terminal is -12 V DC with respect to the "virtual ground." But you could also move the "virtual ground" toward either the positive or negative terminal, producing for example +20 V DC and -4 VDC power supply terminals with respect to the "virtual ground." This could be advantageous if you need mostly a positive supply (to run a motor perhaps), but also need signals that reach zero potential or go slightly negative with respect to the "virtual ground."
When you create a "virtual ground" to allow a single power supply to appear as if it is two power supplies of opposite polarities, there are limitations that you don't encounter with two real power supplies of opposite polarities. The most obvious is source impedance. Two real power supplies will each have very low source impedance, meaning you can pretty much draw whatever current you want from them (up to their maximum current rating) without any effect on their output voltage. It is difficult to create a "virtual ground" that shares this property with respect to its positive and negative terminals on a single power supply. It may be worth the trouble of doing so if you only need, say, a small negative power supply to power up a transducer or something and have only a humongous positive power supply (a car battery for example) available to power everything. But there is always more than one way to do anything. Instead of creating a "virtual ground" you could instead build a charge-pump that converts a positive voltage to a negative voltage. Easy enough to do with modern integrated circuits and power transistors; not so much thirty years ago.
