Most people are accustomed to using unbalanced audio lines, from connecting their stereo and television equipment. It has one wire to carry the signal, and a ground wire (typically, it's wrapped around the signal wire, as a “shield”).
A balanced line has two signal wires, the second signal is an inverse of the first one, and they're used together. They may have a shield around them, but it's not a part of the signal path, it's just there for noise reduction. Equipment accept these two opposing signals to produce the desired one (audio, video, radio, or various other types of signals). But any outside noise picked up by the cable will be the same polarity across both wires, and will get cancelled out.
To demonstrate that mathematically, presume that (at a particular moment in time) down one audio wire is positive 2 volts of electricity, while the other wire carries negative 2 volts of electricity. These are our balanced audio signals. They are combined differentially (the signal from one wire minus the signal from the other wire), and we get 4 volts of signal, in total. This is the audio signal that we want.
(+2) - (-2) = +4
At the same time, the wiring has been placed next to something that induces 3 volts of noise (unwanted signal) into the wiring. Pretty much the same voltage, and polarity, will be induced into both wires at the same time (with only a very slight difference between them, because both wires aren't precisely identical, nor precisely the same distance from the noise source). The input stage combines the signals from the wires differentially (as I've just described, above—the signal from one wire minus the signal from the other wire). This time we get no output, because the noise signal will be subtracted from itself. The noise signal is virtually eliminated.
(+3) - (+3) = 0
Compare that to the unbalanced line, where a fair amount of the noise will get through the shield, straight into the one signal wire, and there being no other way to eliminate it.
These examples are a bit simplified, for the sake of a simple explanation. It's easier to show how balanced audio works with differential signals to produce the desired audio audio, and common (noise) signals cancel out, using text-book example values. But in real life, the in-phase and out-of-phase signal lines may not actually have equal voltages on them, depending on the actual circuit design (though unequal voltages can be a problem with some circuit designs). And the cancellation of common noise across both lines is mostly dependent on the impedances of both signal lines being the same (in balance with each other), across the entire circuit (the output stage, the cabling, and the following input stage). Usually balanced signal wires will be twisted around each other (twisted pair) inside the cable jacket (and inside equipment), this helps to ensure that any external noise that manages to get into both wires, will get into both wires more equally, and that both wire's electrical characteristics will be very similar.
How effective a balanced circuit is at removing common noise is its CMRR (Common Mode Rejection Ratio). The ratio is a comparison of the amount of signal outputted when a signal is applied normally (differentially) to when it's applied as a common signal to both phases, with the result stated in decibels. Typical circuitry may have somewhere around a 70 to 90 dB CMRR, which is very large reduction in noise (90 dB being approximately 32000:1, so that approximately 3 volts of injected noise gets reduced to about 94 microvolts). Compare that to unbalanced wiring, where 3 volts of noise getting onto the signal wire will mean 3 volts of noise on top of your audio signal, and the noise may well be stronger than the wanted audio signal.
At the most basic level, you only need to connect the two signal wires between the equipment. They're the only part of the wiring that carries the signal. The shield, where present, is there to try and prevent external noise getting into the signal wires, or prevent the signal wires radiating noise out. Normally, it'd connect to the chassis bodywork, not the signal ground, though some equipment join the two together, one way or another.
Usually, you'll connect equipment together with an XLR or ¼ inch TRS jack, which will join the two signal wires, and the ground wiring between the equipment. But you may modify that if you have to overcome a problem.
The shield, or ground wire, isn't part of the signal, but it may be necessary to connect your equipment's grounds together to reduce a noise problem. On the other hand, it may be necessary to disconnect the ground wiring between the equipment to get rid of a hum. You'd leave the shield grounded at one end, so the cabling is still covered by a grounded shield. You may need to experiment which end to lift off.
When there's multiple connections between equipment, such as all the wiring between an audio mixer and a stereo recorder, it's usual to connect the left and right grounds together on one piece of the equipment, and disconnect one of the grounds where it connects to the other piece of equipment. And if you have inputs and outputs connecting, too, you might lift the other cable's grounds off at the same side.
e.g. A tape deck would have all the wiring shields connected to its inputs and outputs. The mixer might only have the shields connected on the left in and left out. Or, perhaps, only on its left input.
This is done to reduce noise. On some equipment you can hear it make a significant change, if you listen while you change the ground wiring around. Ultimately, you connect and disconnect grounds to get the quietest background noise levels that you can manage, not stick to a strict rule of always connecting this and always disconnecting that. And you need to listen to the effect it has on all the equipment, not just the device you're currently adjusting. e.g. You might make a tape deck quieter by connecting all its grounds, but make the mixer noisier, all the time, by doing so.
Way back in the olden days, when dinosaurs roamed the recording studios, most balanced connections were made through audio transformers. This meant that the real connection between different equipment was a magnetic coupling between the two halves of the transformer. There wasn't any direct electrical connection, unless you joined ground wires, too. This gave you a lot of isolation, making it very easy to connect lots of equipment together, without one thing causing problems to another.
Nowadays, it's a rarity to find audio coupling transformers in the input or output stages, and you can get ground loops from the signal wiring, no matter what you try to do by changing the external ground connections. Since, ultimately, somewhere inside the equipment, the input and output stages are connected to ground. With equipment, like that, you may have to insert your own audio transformers into the signal path, to break an earth loop. Particularly if connecting equipment together that are in different rooms, or buildings.
That's another of those questions that doesn't have one always-true answer. The simplest way is to use an audio-coupling transformer. It doesn't need power, it breaks earth loops, you don't unbalance a balanced line, you can match impedances, and you shouldn't have to worry about what audio level is going into it. While some will argue that a transformer will colour the sound, I'll counter that since I've never seen the audio mixer operator leave all the equalisation flat, the sound is going to be coloured, anyway.
The only problem I find with using transformers, is that they can pick up hum from being close to a noise source. Though, on the other hand, they can eliminate hum picked up by the cable leading up to them.
The alternatives are:
Active direct injection boxes, which have an amplifier in them that could be driven into distortion by strong signals, or introduce their own noises to weak signals, so you'll have to have attenuation controls on them. They need a power supply, they can't easily break ground loops, and they can still colour the signal.
Or to unbalance the line by directly connecting the signal lines together, the grounds together, and the out-of-phase signal line to ground. This risks all sorts of noise problems, and you can only safely connect an unbalanced source to a balanced input that doesn't supply phantom power. And it's a very bad idea to try grounding half of a transistorised balanced output stage, to connect to an unbalanced input, as you can destroy the output stage. Connecting the out-of-phase line to a dummy load to ground, rather than directly to ground, is a better approach. Though you can still get noise problems, thanks to impedance mismatching between the source and input stages.
Using a transformer is the simplest and easiest solution.
Usually, you'll find you can get away with simply running an unbalanced line between the microphone and the equipment, if it's just a few metres long. And you'll find lots of cheap dynamic microphones come with a cable wired that way (a balanced XLR to fit into the microphone body, and a five metre cable leading to an unbalanced ¼ inch jack to plug into an amplifier or mixer). This often works well, until you find yourself in a hostile environment (e.g. there's a lighting dimmer nearby), or you have to use a longer cable (which will pick up much more ambient noise than a shorter lead would).
The simplest solution, again, is to use an audio transformer. Place it as close as you can to your input stage, it's (now) giving you a balanced input at your equipment, so that your long microphone cable (now) works as a balanced line. Balanced lines are much better at rejecting noise and interference than unbalanced lines. In some cases, they can reliably reject noise that's actually stronger than your microphone audio signal. Probably not entirely, but enough that it's no-longer a big problem.
You can buy bare audio transformers, for building into your own equipment (so you could convert the input stages of a cheap mixer, if there was space inside the box), or for making up your own adaptors. You can buy transformers in boxes, so you can wall-mount them next to fixed installations. And you can buy them built into plug and socket adaptors, for field work and temporary problem solving. If you're going shopping for any of these, what you're looking for could be called any one of these names: DI boxes, direct inject boxes, audio coupling transformers, impedance matching transformers, or microphone transformers.
For schools, churches, etc., having to make do with simple unbalanced input audio mixers, and having hum problems from long leads going all over the place, it'll save you a lot of grief by putting audio transformers in the signal path to convert your unbalanced inputs to balanced lines. And you'll find using well-constructed XLR leads to be much more reliable than ¼ inch jacks, especially if you have to join cables together. But, if you're faced with having to buy quite a few transformers, especially good quality ones, you may find it cheaper to buy a new mixer.
This is another of those questions commonly faced by people running audio equipment on the cheap in churches and schools. And the answer is, it depends…
If you don't have any noise problems, it isn't worth it. If the cabling is fairly short, it probably isn't worth it. If the cabling is long, or goes through a hostile environment, then it probably is. And if you have hum or noise problems, it's definitely worth seeing if you can get rid of the noise in this manner.
At your unbalanced equipment, connect a transformer as close as reasonably possible, this will (now) give you a balanced audio connection. You do the same at both the source and input stage (e.g. you put a transformer close to the output socket of an electronic piano, and likewise at the input socket of your audio mixer). And you run balanced audio cable between the two of them.
At the input stage, you may need to experiment between connecting the long cable's shield to the centre tap of the input transformer, the ground of the unbalanced input socket, or both. But, typically, it'd connect to the mixer's ground (easily done through the unbalanced input socket). DI boxes usually come with one or two switches for that purpose (earth lifting).
You get two benefits from converting, like this, to a balanced line: Reduction of noise picked up by long cabling. And using a transformer gives you a way to eliminate hum caused by earth loops between the equipment.