In an unbalanced cable, there's a single inner core that carries the signal while the outer screen also doubles as the signal return path. Any interference that results in induced currents flowing through the screen conductor will cause these currents to be added to the wanted signal, so some interference problems are still possible, especially with long cable lengths or where there are nearby sources of strong interference. Screened cable offers very little protection against induced hum, as the cable is actually acting like a single-turn transformer, coupling energy from other mains conductors and transformers in the vicinity.
In a balanced cable, there are two inner conductors, often known as hot and cold. The screen is grounded, as before, but this time the screen is not part of the signal path.
Balanced equipment is designed so that its hot and cold outputs carry the same signal but with the cold signal phase inverted with respect to the hot signal. At the receiving end, the balanced input stage re-inverts the cold signal and adds it to the hot signal, thus restoring the original signal.
Another type of cable known as Star Quad was developed to improve the immunity to interference, these kind of cables are used mainly in live sound applications where long cable runs are commonplace, but it is demonstrably more effective than conventional balanced cable.
So if you use unbalanced cables keep them short and if it's possible far from transformers and power cables.
