Its a bit like comparing single-coil pickups to humbuckers.

An unbalanced line is usually a two-conductor cable, with a 'hot' and a 'neutral' wire. The signal is carried as a voltage difference between the two wires. Since the neutral wire is connected to the system ground, any noise induced on the cable looks just like part of the signal, as far as the circuit on the receiving end of the cable is concerned.

A balanced line has a + line, a - line, and a Common line. Most interferance will be coupled onto the cable in 'common mode' (all of the lines get the same signal). The circuitry at the receiving end of the cable uses only the differential between the + and - signal, relative to the Common line, so the common-mode voltage is ignored. This results in a 'quieter' signal path having a better signal-to-noise ratio.