When depletion zones are thick, you can boost the voltage on the base electrode. The voltage at this electrode is directly controlled by the input current. When the input current is flowing, the base electrode has a relative positive charge, so it draws electrons toward it from the emitter. This frees up some of the holes, which shrinks the depletion zones. As the depletion zones are reduced, charge can move from the emitter to the collector more easily -- the transistor becomes more conductive. The size of the depletion zones, and therefore the conductivity of the transistor, is determined by the voltage at the base electrode. In this way, the fluctuating input current at the base electrode varies the current output at the collector electrode. This output drives the speaker.
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A single transistor like this represents one "stage" of an amplifier. A typical amplifier will have several boosting stages, with the final stage driving the speaker.
In a small amplifier -- the amplifier in a speaker phone, for example -- the final stage might produce only half a watt of power. In a home stereo amplifier, the final stage might produce hundreds of watts. The amplifiers used in outdoor concerts can produce thousands of watts.
The goal of a good amplifier is to cause as little distortion as possible. The final signal driving the speakers should mimic the original input signal as closely as possible, even though it has been boosted several times.
This basic approach can be used to amplify all kinds of things, not just audio signals. Anything that can be carried by an electrical current -- radio and video signals, for example -- can be amplified by similar means. Audio amplifiers seem to catch people's attention more than anything else, however. Sound enthusiasts are fascinated with variations in design that affect power rating, impedance and fidelity, among other specifications.
For much more information on amplifiers, check out the links below.