Role Of Emitter (By Pass) Capacitor
Usually, this capacitor is in parallel with a resistor:



In a common-emitter amplifier, any impedance between the emitter and ground (\$R_e\$ and \$C_e\$) serves to reduce the gain of the amplifier: it is a form of negative feedback. By increasing the negative feedback and decreasing the gain, we can make variations in transistors less significant. Perhaps most relevant to this point, by adding \$R_e\$ we make the bias current more dependent on the resistors (which are easy to control) and less dependent on the gain of the transistor (which varies over a large range, even among transistors of the same model).

But what if we still want high gain? Because a capacitor presents an impedance that decreases with frequency, putting \$C_e\$ in parallel with \$R_e\$ serves to decrease the negative feedback, and thus increase the gain, at high frequencies. Effectively, high-frequency signals can bypass the emitter resistor, through the capacitor. Yet, to DC, the capacitor appears as an open circuit, so adding the capacitor does not affect the DC bias current. Thus, if we are interested in amplifying AC signals only, this capacitor allows us to have a stable DC bias current while maintaining high gain for our signals of interest.