the small-signal model
A transistor is a nonlinear device — its collector current depends exponentially on the base-emitter voltage, which is a nightmare to do algebra with. The small-signal model is the engineer's escape: if the signal is SMALL compared with the DC bias, you can pretend the transistor behaves linearly around its Q-point and replace it with a tidy little circuit of resistors and a current source. It is the act of zooming in so far on a curve that the tiny piece you are using looks like a straight line.
The standard version is the hybrid-pi model. After you have found the Q-point (the DC bias), you redraw the transistor for AC as just two elements: a resistor from base to emitter called r-pi (representing the input the base presents), and a current source from collector to emitter whose value is gm times v-be — that is, the transconductance gm multiplied by the small signal voltage across the base-emitter. The key parameters all come from the Q-point current Ic: the transconductance is gm = Ic/VT where VT is the thermal voltage of about 25 mV, the input resistance is r-pi = beta/gm, and the small internal emitter resistance is re = VT/Ie, about 25 ohm at 1 mA. With these three numbers you can compute any stage's gain and impedance with nothing harder than Ohm's law.
This is how you actually design and predict amplifiers. Want the gain of a common-emitter stage? It is just -gm times Rc. Want its input resistance? r-pi in parallel with the bias resistors. The whole zoo of configurations collapses into the same little model with the terminals connected differently. The honest boundary is right there in the name: SMALL signal. The moment the signal is large enough that the transistor's curve bends appreciably over the swing — or it nears cutoff or saturation — the linear model breaks down and you get distortion the model cannot predict. The model is a local approximation, true only near the Q-point you linearized around.
Bias a transistor at Ic = 1 mA. Then gm = 1 mA / 25 mV = 40 millisiemens, re = 25 ohm, and with beta = 100, r-pi = 100/0.04 = 2.5 kohm. Drop these into a common-emitter stage with Rc = 5 kohm and the predicted gain is -gm times Rc = -0.04 times 5000 = -200, no transistor curves required.
Three numbers from the Q-point (gm, re, r-pi) let you predict gain with Ohm's law.
The small-signal model is only valid AROUND a Q-point you must establish first with DC analysis. It says nothing about biasing, clipping, or large signals — those need the full nonlinear picture.