Three types based on common electrode:

The common electrode is shared by input and output circuits and is generally grounded.

Emitter is common to both input and output signals.
Input signal applied between base and emitter.
Output taken between collector and emitter.
Produces highest current and power gain among all BJT configurations.
Input impedance: LOW (due to forward-biased PN-junction).
Output impedance: HIGH (due to reverse-biased PN-junction).
Used when large current gain is needed, bias stabilization, audio frequency applications.

Describes relationship between input current ($I_B$) and input voltage ($V_{BE}$).

$I_B$ on y-axis, $V_{BE}$ on x-axis.
To determine, $V_{CE}$ is kept constant (e.g., 0V, 10V, 20V) and $V_{BE}$ is increased.
For each $V_{BE}$, corresponding $I_B$ is recorded.
When $V_{CE}=0V$ and EBJ is forward biased, it acts like a normal p-n diode.
After 0.7V, a small increase in $V_{BE}$ rapidly increases $I_B$.
In CE, $I_B$ is small due to lightly doped, narrow base.
In CB, input current ($I_E$) is large due to heavily doped, wide emitter.
Input current in CE is in microamperes ($\mu A$), in CB in milliamperes ($mA$).
EBJ is forward biased, CBJ is reverse biased. Depletion region at EBJ is small, at CBJ is large.
Increasing $V_{CE}$ further increases depletion region width.
Base region is lightly doped, so depletion region penetrates more into base. This reduces input current ($I_B$).
For higher $V_{CE}$, the curve shifts right, increasing cut-in voltage above 0.7V.

Describes relationship between output current ($I_C$) and output voltage ($V_{CE}$).

$I_C$ on y-axis, $V_{CE}$ on x-axis.
To determine, input current ($I_B$) is kept constant (e.g., $0 \mu A, 20 \mu A, \dots$) and $V_{CE}$ is increased.
For each $V_{CE}$, corresponding $I_C$ is recorded.

Cut-off region: When $I_B = 0 \mu A$, both junctions are reverse biased.
Active region: Emitter-base junction forward biased, collector-base junction reverse biased. This is where amplification occurs.
Saturation region: When $V_{CE}$ is reduced to a small value (e.g., 0.2V), CBJ becomes forward biased. Both junctions forward biased. Small increase in $V_{CE}$ rapidly increases $I_C$.

Ratio of output current (collector) to input current (base). $$ \beta = \frac{I_C}{I_B} $$
High in CE configuration, used for current amplification.

Ratio of change in input (base) voltage to change in output (collector) voltage. $$ h_{re} = \frac{\Delta V_{BE}}{\Delta V_{CE}} \quad \text{at constant } I_B $$

Ratio of change in input voltage ($V_{BE}$) to change in input current ($I_B$) at constant output voltage ($V_{CE}$). $$ r_i = \frac{\Delta V_{BE}}{\Delta I_B} \quad \text{at constant } V_{CE} $$
Very low in CE configuration.