An unbalanced line is a cable that is not symmetrical from an impedence perspective. If you can remember the ancient days when everyone had a television antenna on their roof, folks had a flat cable that ran from the antenna to the back of the television. Even if you had rabbit ears for an antenna, there was a balanced line of two parallel wires that went to the telly.

Eventually the more efficient and less noise-prone coaxial cable came into use. Because cable television broadcast their signals near the frequencies used by aircraft, they had to make sure that their signals didn't cause interference. One of the methods they chose was to switch to coaxial cables, also known as an unbalanced line, because the ground and shield helped to contain the broadcast signal.

The reason it's called an unbalanced line is because the two transmission wires have an unequal impedence as compared to ground. The old television twin lead cables were balanced because both wires were exactly the same impedence. 

For ham radio operators, some antennas prefer to be fed with a balanced line. The problem is that you can get a lot of interference and standing waves of high voltage on balanced lines. Most radio operators end up using an unbalanced line to feed a balun, which is an adapter between a balance and unbalanced transmission line.

Iron Noder 2017