the emitter bypass capacitor

The emitter bypass capacitor solves a dilemma. You need an emitter resistor for stable DC biasing (it sets the Q-point and stops thermal runaway), but that same resistor crushes your AC voltage gain down to -Rc/Re. Can you have stable bias AND high gain at once? Yes — bridge the emitter resistor with a capacitor, and you get the best of both: rock-solid DC bias, full AC gain. The capacitor is a frequency-dependent bypass, a back road that only opens for the signal.

Here is the mechanism. A capacitor blocks DC entirely but conducts AC, more easily the higher the frequency (its reactance is 1/(2 times pi times f times C)). So to the steady DC bias current, the capacitor is invisible — the emitter resistor is fully in circuit, holding the Q-point firm. But to the AC signal, the capacitor is a near short-circuit that bypasses the emitter resistor, effectively grounding the emitter for signals. With the emitter AC-grounded, the gain leaps from -Rc/Re back up to the full -gm times Rc. The capacitor must be large enough that its reactance is much smaller than the internal emitter resistance re at the lowest frequency of interest — for audio down to 20 Hz with re around 25 ohm, you typically need tens to hundreds of microfarads, which is why it is usually a bulky electrolytic.

Why it matters and where it bites: this single capacitor lets one stage deliver both DC stability and high gain, and you will see it on the emitter of almost every discrete high-gain stage. But it sets the low-frequency limit of the amplifier — below the frequency where the capacitor stops looking like a short, the gain rolls off and the bass disappears. There is also an honest tradeoff: a fully bypassed stage has high but BETA-dependent, less linear gain, whereas leaving a small un-bypassed emitter resistor in series (partial bypass) sacrifices some gain for much better linearity and a defined input resistance. Designers often split the emitter resistor into a bypassed part and an un-bypassed part to tune that balance.