What does it mean when an input or output is balanced?
This topic is commonly misunderstood, and Joel's answer isn't quite correct.
Transmitting "both the signal and its opposite along two wires" is called differential signaling. This is used to minimize emissions from a cable into other circuitry, because the equal and opposite electric and magnetic fields cancel out at a distance from the wire.
A "balanced line", on the other hand, means that the sender and receiver both have the same impedances connected to each of two wires (the source impedance for both lines is the same and the load impedance for both lines is the same). This way, any interference into the cable produces the same voltage on each line, and can therefore be canceled by a differential amplifier in the receiver.
Usually these are both used at the same time, but they don't have to be. The so-called "impedance balanced" outputs on some mixers are an example of a circuit that is balanced but not differential.
SIGNAL SYMMETRY HAS ABSOLUTELY NOTHING TO DO WITH NOISE REJECTION — IMPEDANCE IS WHAT MATTERS!
It's really the cable that's balanced.
Imagine a long cable transmitting a weak signal (for example, a very low voltage signal from a microphone). As the signal travels down the cable, ambient noise might distort the signal.
In a balanced cable, instead of just transmitting the signal by itself, you transmit both the signal and its opposite along two wires, usually twisted together.
If there is any interference, the interference will apply equally to both wires.
That allows it to be cancelled out at the receiving end automatically.
Imagine that the signal is +5 and the noise is +1.
An unbalanced cable would transmit +5, but the noise would cause a +6 to arrive.
A balanced cable would transmit -5 and +5; the noise would cause a -4 and +6 to arrive. Now the receiving apparatus can figure out that the original signal must have been +5.
In practice, you'll recognize balanced cables because they use three conductors instead of two, for example, a TRS 1/4" jack:
Or an XLR jack:
It's an interconnect method that lets you transmit signals over very long lengths of wire without having large amounts of extraneous noise injected in to the signal. The signal is duplicated on to two wires and the input impedance at the received is the carefully matched for both signals. This insures that noise injected during the journey is done so in equal amounts on both signals so it can be detected and rejected by the receiver. This is important for relatively weak signals (say from a microphone) where even minute amounts of RF noise picked up on the line are, relative to the signal level, fairly large.
This wikipedia entry describes the different signaling methods in balanced transmission design, in detail.