What is biasing?

In very simplistic terms, tubes operate with a voltage applied to them to get the best performance in the circuit. The amount of voltage that needs to be applied to drive the tube optimally is known as the bias voltage. Some modern tube amps monitor and adjust the bias automatically, but traditionally there is a small adjustment pot which is used to change the bias voltage. Not enough bias voltage and the tube runs "cold," too much bias voltage and the tube will distort early (in a less musical way) and the stress of the increased voltage reduces the life of the tube.

For amps that allow for user adjustment of the bias, there then needs to be some way to check when the correct biasing is achieved: either via a voltmeter (built into the amp or supplied externally by the user) or an LED on the amp somewhere.
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Hey, I got it--- It was clear as mud.

"Bias" in any context means something which is a steady value, on top of which some variation occurs. For a vacuum tube, bias current means the current that flows through the tube (from the power supply to ground via a beam of electrons in the tube) when the control signal (applied to the grid of the tube) is zero. This background current flow is set at a level where the tube works best by application of a bias voltage (DC) to the control grid. The audio signal varies the voltage on the grid, which varies the current through the tube, which also flows through the primary winding of the audio output transformer, which generates a voltage in the secondary of the transformer, which goes to your speakers...whew! Adjusting the bias is a ritual solemnly performed by some audiophiles. There are also some amplifiers that automatically adjust their bias, but this takes the fun out of life.

I like Rushton's answer. Simplistic works best for me when it comes to hot and cold running electrons. Eldartford's answer was far too correct! (Actually very helpful, thanks.)

We just biased our new tube amp for the first time, manually. Really, really exciting moment. There's nothing like dragging 65 pounds of iron out of the rack just so you can get to it with a multimeter. Auto-biasing must be like auto-return tonearms, keeps you from truly bonding with your stereo.

One possibly useful tidbit: if the amp has more than one output tube per channel there may be a facility for balancing the tubes against each other as well as biasing the overall DC voltage level. Unbalanced tube voltages on the same channel apparently feeds DC into the output trannies, which tends to saturate them. (Eldartford can correct that if I messed anything up.)