Balanced connections allow long cable lengths with reduced susceptibility to noise.
Detailed Explanation
Balanced connections use a 2-core cable. [footnote=1]NB: The term 2-core is actually shorthand for '2 cores plus shield'. So for example under this notation, a guitar cable would be a single-core cable! It is assumed that basically all pro audio connections have a shielding wire. For example the wire used in a home stereo system has two cables running side-by-side. Here the return wire also has to act as a shield. Whereas in a professional audio cable, the shield is braided around the outside of the core, or signal, cables. This offers improved noise reduction[/footnote]
The signal at source within the microphone is split into 2 with these 2 cables carrying the audio signal. Circuitry within the microphone puts the cold (-) wire 180̊ out of phase with the hot (+) signal.
At the other end of the cable run, the two signals are recombined by a differential amplifier.
Noise introduced to the hot wire will also be introduced to the cold wire. When the two signals are recombined the phases are flipped - the initial audio signal is put back in phase on both wires, and the introduced noise is now out of phase, cancelling itself out.
Applications in Pro Audio
Simply put there are few places where a balanced signal is not[/] required in pro audio. Cable runs from microphones to mixers and converters, within patchbays and between outboard gear, are not only susceptible to noise but the noise may then be amplified over, say, 16 channels, making balanced connections essential.
Due to the microphone signal itself being so low level any noise introduced would be at a high relative level to the signal. This means that 100% of the time, your microphone cables need to be balanced, and XLR connections are generally used because of their durability and the built in locking mechanism ensuring that they aren't accidentally pulled out while phantom power is applied to the microphone, causing potential damage. This same thinking is applied to most outboard gear, with mixers and patchbays using TRS 1/4" jacks due to their smaller diameter.
Most audio gear features balanced connections, up to and including long signal runs from mixing desks to PAs. Once you reach amplified volumes, speaker cables tend to be unbalanced, and some line level equipment, notably keyboards/synths, are limited unbalanced outputs. This is generally a cost-saving measure and part of what makes pro audio an expensive hobby is the need to constantly balance and unbalance signal chains.
Internal Balancing
The signal inside most pro audio products (compressors, EQs, preamps etc) is internally converted to an unbalanced signal for processing. The units generally feature much improved internal shielding and very short signal paths, making the requirements for balancing long cable runs to prevent noise moot. In addition this internal processing is performed at line level or very hot line levels, making signal-to-noise a lower consideration.
The balanced input signal is converted to an unbalanced signal via transformer or op-amp (hence: transformer-balanced preamp inputs) and after internal processing, converted back. We refer to the input and output signals as being isolated from one another due to the differential amps at either end of the circuit.
However some pro audio gear offers a fully balanced signal path internally, mirroring the circuit path for the (-) and (+) signals. This is obviously expensive to do but can result in a different quality to the sound, and better signal-to-noise ratio.