You guys are bouncing around all of this, but without the true specifics.
Class A means an operating mode where the biased current is sufficient to exceed the load. Bias means the amount of current or voltage biased above and beyond the steady-state, no-load amplifier circuit without outside interference.
A great example of this is a demo amp that Nelson Pass once built. It was biased by using two incandescent light bulbs. IOW, the amplifier circuit was forced (biased) to run at a much higher current than it "wanted to" without a load. The current from burning the two light bulbs was routed through the current sensing legs of the amplifier's circuit, thus producing a bias that pushed the amplifier circuit into Class A operation.
Pete Millett did the very same thing on this forum years ago with the Starving Student Headphone amplifier. He routed the tubes' heater circuits through the current sensing legs of the output MOSFETs (it was a tube hybrid design), thus biasing the amplifier into Class A operation.
Why would one do this? To reduce noise and distortion. Every other amplifier topology uses some sort of switching, most often in the transistors themselves. Transistors turn on and off in response to a load. In audio - especially headphones - we are interested in hearing the very smallest of signals. We don't want to tolerate the sources of any other noise, such as switching in a transistor or MOSFET.
This is also why linear-regulated power supplies are preferred in high-fidelity audio: because they don't use switching. Modern switching power supplies claim often claim no switching noise in the audio band, but more often than not, it's not true.
Back to Class A - all of this is contingent on the load. If the current demanded by the load exceeds the biased current in the amplifier, then it's no longer running in Class A. In most cases however, the circuit is designed to run in Class A, so the bias amount selected for operation is selected to exceed any possible reasonable load that the amplifier might encounter.
Bottom line, "bias" is the amount of current increase over no-load conditions. If that current bias is sufficient such that the load never exceeds that current, then the amplifier is operating in Class A and there is no switching noise produced (assuming the power supply is not a switching power supply).
P.S. Why are there other amplifier Class operating modes? Quite simply, once you exceed the power levels of headphones, you are talking about massive amounts of heat that has to be rejected. Even a headphone amplifier that is biased to a couple of hundred milliamps will need massive heat sinks. Ever seen a Gilmore Dynahi? They're usually two massive heat sinks that basically form the two entire sides of the amplifier enclosure. When you get to speakers, it almost becomes impossibly uneconomical to run Class A - thus the other amplifier Class operating modes.