In this diagram the DC power sources for a linear amplifier are shown:
We see there are two separate batteries connected to it, and then to ground on opposite ends.
Why must there be two separate power sources connected to a linear amplifier? Why not just connect it to a single power source with a specified voltage?
A single positive power source (referenced to 0 V) is fine if you only require positive-going output voltages, but if you require your output voltage to swing below ground (0 V) then you need a negative power rail.
You can't have negative-going output voltages without a negative power rail. Having two separate power sources allows the creation of the 0 V reference between them to which both input and output are referenced.
Op amps have an output voltage swing which is limited by the power supply inputs -- the output can at most swing up to the positive supply ("rail") and down to the negative supply. Many op amps can actually only swing to within a few volts of the rails. The datasheet for the (in)famous 741 op amp, for example, shows this limitation (the value on the far right is the "typical" value, and the value just to its left is the specified minimum):
There are some "rail-to-rail" op amps which are capable of swinging all the way to either power supply voltage, but in general an op amp with only a single supply will not be able to output 0 V or below.
Similarly, op amps also have an input common mode range that is limited by the power supply inputs. Again, many op amps require the input common mode voltage to be a few volts away from both power supply voltages but there are also some "rail-to-rail" input op amps. The 741's datasheet gives its input voltage range with +/-15 V supplies as:
Some op amps are marketed as "single-supply", in which case the input common mode range and output voltage swing extend to the negative supply -- that allows you to power such an op amp with only a single power supply.
An op amp that is not rail-to-rail or single supply will not operate properly with only a single supply if you have an input signal that swings too close to (or below) ground (which in that case is the op amp's negative supply) and/or you need the output to swing down to ground or below.
What they’re showing is an idealized dual-supply, or bipolar amplifier. Bipolar amplifiers have the ability to both accept and output signals that swing above and below the ground reference. This is made possible by the dual supply, which allows both positive and negative biasing. This is the classic style of op-amp powering, going back to the 1960s and parts like the uA709 and uA741, which are designed to run on +/-15V supplies.
(Note: confusingly, bipolar power is a separate concept from bipolar transistors.)
The alternative is the single-ended or single-supply amplifier. These will usually use GND as the negative amplifier supply. Thus, they limit their input and output swings to be between GND and the positive supply as there is only positive bias available. The LM324 is a well-known example of a single-supply op-amp, although it too can use a bipolar supply.
(Second Note: if you’re talking about real op-amps, review their data sheets carefully and observe their limits: their I/O voltage ranges will usually be less than the supply rails. On the other hand, also note that there are specialty op-amps that can accept wider input voltages, e.g., op-amps designed for high-side current sensing.)
Which kind of powering you would choose depends on kinds of signals you’re processing and your desire for low system cost:
If your signals are always above ground you would choose single-ended for lower cost. This is common for signal buffers to microcontrollers for example.
If your signals are bipolar AC (like audio), you can still use single-ended, with the addition of an artificial midpoint bias reference and AC coupling. Most portable audio electronics are built this way.
If your signals are very low frequency or DC (and thus can’t use AC coupling) and swing above and below ground, using bipolar supplies solves a lot of problems, at the expense of a separate negative supply. You often see this in instrument signal blocks, in high-end audio, and sometimes for high-power servo drives.
Finally, if your signals have a large swing, using bipolar allows for handling a greater voltage range. A uA741 powered with +/-15V can easily handle +/-10V signals.
That all said, textbooks tend to assume bipolar amplifier supplies because it simplifies input and feedback network analysis: they don’t have to consider the influence of an artificial midpoint bias.
One reason to use a dual supply on an amplifier is to eliminate the need for coupling capacitors.
In an audio power amp for example, the AC signal needs to swing around a fixed DC voltage. With a single supply that voltage will need to be half the supply voltage so the signal can swing an equal amount above and below it. If you were to attach a speaker directly to that, you would have half the supply voltage across the voice coil and it would either burn out or be deflected towards one extreme of it's travel. To eliminate that they put a large electrolytic capacitor in series with the speaker to block the DC.
If you use a dual supply, you can make the DC voltage at the output be 0 V with respect to ground. The signal can still swing above and below that because there is a positive and negative supply, but there will be no DC on the output so you can connect the speaker directly to it without a capacitor.
You can see in your schematic that the load resistance is connected directly to the output, if there was a single supply either the amplifier would need to be biased so that it's at ground with no signal and can only go positive, or there would be DC on the load.
This is more related to the electronic construction of the input terminals from OPAMPS: a differential pair
If you carefully analyse the input stage from the venerable and rather horrendous LM741, you should be able to distinguish a differential pair connected to a current source below such input:
Symmetric supply voltage is needed to allow (as others have pointed out) full excursion of a signal entering the two terminals of the OPAMP. Basically, you want to establish a setpoint so that your signal is referenced right at a virtual zero allowing output swing between approximately Vcc and Vee (applicable only in true Rail-to-rail devices, and not quite, though). When both terminals are tied together, for instance, the output voltage should be zero as internally the OPAMP is computing the difference between the potentials at each terminal:
In Open loop:
$$V_o = A_v(V_+ - V_-)$$
where \$A_v\$ is the open-loop gain.
Since this is practically not possible offset null is added in some configurations to overcome input offset voltage
