From: John Doty (jpd@space.mit.edu)
Date: August 24, 1995
Original source: Usenet's rec.radio.shortwave

The purpose of a balun is to block common mode waves (current flowing in the same direction on both conductors of a transmission line) and pass differential mode waves (current flowing in opposite directions on the conductors). While some baluns also transform impedances, others don't: "1:1" baluns are very common.
There are many reasons to want to block common mode waves, but for receiving the major one is that noise from your house wiring can easily reach your antenna by travelling up your feedline in common mode. It isn't necessary to use a balun to block the common mode, however: with coaxial cable, grounding the shield short circuits the common mode.
One source of confusion here is that there are a couple of companies that make devices they call "Magnetic Longwire Baluns", which are not baluns at all (they don't block common mode", but are simply impedance transformers.
There's plenty of theory on the impedance of wire antennas: they are very predictable. Basically, in the complex plane, the impedance is a spiral that approaches its central point as frequency increases. The central point is the "characteristic impedance" of the wire, typically 400-700 ohms depending on the wire diameter and height above ground. While matching the cable to this point is never a perfect match, it is almost always good enough for receiving frequencies from longwave through shortwave, because natural noise levels are so high that natural noise picked up by the antenna dominates receiver noise even if some signal is lost due to the imperfect match. For receiving VHF and above (where the natural noise is very low), optimum matching is more important, and it is also more important when you are transmitting at any frequency.

Is a balun useless on a long wire antenna?

No. First of all, a true balun can help block common mode noise. Secondly, we're really discussing matching transformers here. A transformer that matches a coaxial line to about 500 ohms attached to a wire antenna at least 15 meters long will transmit enough signal to allow most receivers to reach the natural noise limit from 100 kHz to 30 MHz. A better match yields stronger signals, but does not improve the signal to noise ratio. With many receivers, stronger signals are actually undesirable, due to difficulties with overloading.