A BJT biased in forward-active operation can use a small base-emitter voltage variation to control a larger collector-current variation.

Around the operating point,

$$i_c=g_m v_{be},$$

with

$$g_m\approx \frac{I_C}{V_T}.$$

In a basic common-emitter stage with collector resistance $R_C$ and sufficiently large transistor output resistance,

$$v_o\approx -g_mR_Cv_{be}.$$

The output is inverted because increasing collector current increases the voltage drop across $R_C$ and lowers the collector voltage.

The DC operating point determines the local transconductance. The small-signal model then predicts gain only for variations small enough that the transistor remains near that operating point.

Large signals can drive the transistor into cutoff or saturation, where the linear gain relation no longer applies.