I am no transformer expert. There has been much made about d.c. as in the article frank provided, but there are other reasons and I have no idea which is most prevalent. Fundamentally they all relate to some sort of effect whereby mechanical vibration is induced by the EM (electromagnetic) field of the transformer. Here are a few off-the-cuff thoughts, which may or may not be true, and in any event I doubt any are worded terribly rigorously, in random order:
1. Might as well acknowledge the d.c. issue first. The idea is that a relatively small d.c. current "uses up" transformer headroom and causes the core to saturate early. I am not at all sure I buy it, but it is not something I have thought a lot about. (In fact, none of this is something I have thought a lot about.) A transformer depends upon a.c. current flow creating an alternating magnetic field that induces current flow in the other winding. Magnetic flux can lead to mechanical movement, and as the core saturates it becomes asymmetric, fields no longer cancel, and buzzing can be induced. D.c. current does not alternate, causing a sort of "offset" that uses up some of the core's dynamic range. Other causes...
2. Asymmetry in the current flow which may come about due to d.c., input (line) imbalance, or load (diodes and filtering) imbalance.
3. Overloading the core (drawing too much power) -- this causes severe hysteresis and subsequent high harmonics and core imbalance.
4. Excessive heat reduces magnetism in most materials and leads to early saturation of the core, which causes harmonics and makes it more likely to buzz.
5. High harmonic content induced by overload, the power rectifiers reflecting back into the transformer, etc.
6. The magnetic and electrical fields are varying 60 times a second and are 90 degrees "out of phase". Sometimes the glue holding the wire (windings) and/or core (typically laminated thin metal pieces built into a thick core) gets old and dry and cracks, or is defective from the factory, allowing wires and/or core to shift during operation and produce a buzzing sound.
7. Those same EM fields cause normal mechanical stress; if you put your hand a hard-working transformer, even if it is silent, you can fell it vibrating. If the transformer is not either isolating (e.g. by rubber bushings) or solidly mounted to a heavy chassis you can hear some mechanical buzzing.
8. An external magnetic field can disrupt transformer operation; I consider this very unlikely but had already typed "8" and didn't want to end on an odd number.
The key point is that a transformer always has electrical and magnetic fields going so the potential for buzzing is always there; well-designed transformers and their mounts reduce it to inaudibility, and poor design and/or high asymmetric loads can make them hum, buzz, whine, sizzle, and generally misbehave.
HTH - Don
p.s. I did not mention ground loops as that is a separate issue. A power cord change should not do anything to the transformer itself, but as JN has noted can change the system ground paths. Most of us think a transformer breaks a ground loop, but in practice the safety ground is typically connected to the chassis and so is the transformer's case and core. Many designs lift the input signal (not power) ground with a low-value resistor right at the input, before the first gain stage, to reduce the chance of incoming signal mixing with the power ground (the resistor makes the return path through the power ground too high in impedance) and thus reducing the ground loop to inaudibility. More or less (JN will probably correct my loose and imprecise verbiage, but I'm tired tonight).
