For each channel, say the "left" channel, balanced amps and preamps have separate sections to amplify the positive half of the signal and the negative half of the signal. This is one way of looking at it, in reality they an amplifier with positive gain and an amplifier with negative gain, each is closely matched to be the inverse of the other. The output is the differnce between these two amps, and that is why in theory balanced amplifiers should have twice the output of single-ended amps.
For the connectors, one wire connects to the + gain, the other to the - gain, and you are and ready to go. So the XLR/Balanced inputs have 3 wires (one for + one for - and one for ground (neutral)) and 3 contacts to carry the signal to the balanced circuitry.
Why is this good? Because any noise that is common to the system will be amplified in the + section and amplified in the - section, and when you take the differnce for your output the noise cancels itself out. This is called "common-mode" noise rejection. This is also why long runs of balanced cable are better than long runs of single ended cable. Any noise picked up equally by the wires in the cable will be canceled out when the amp looks at the difference signal between the 2 cables.
Just because something has balanced connectors does not mean the circuitry is truely balanced with the separate internal amps I just mentioned. Some coponents just convert signal from the balanced cable into a single ended signal with a cheap op-amp and use a traditional type of amp. If you are considering BAT gear that would be truely balanced, as are the Sonic Frontiers preamps.