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have a look at the I-V curves for diodes, especially zeners. they helped me understand - when a Zener is in breakdown, its slope is VERY steep. This means that for a LARGE change in current (vertical), there is a SMALL change in voltage (horizontal), which is why they are good regulators.
Not great amplifier mind you! Diode's have a linear region during their pass from non conducting to conducting, you can bias the diode with a fixed voltage and capacitivly couple an external signal into it. Distortion is horrible though, not particularly useful, though I've heard some crystal radio makers use a bias voltage to increase sensitivity.