We all must be aware about the meaning of the current. Yes, it is rate of flow of charge. Do you know that when an electric circuit having a ohmic resistance is switched on, the current reaches to maximum value almost in zero time and when the circuit is switched off, the current drops to zero from maximum value in a no time. This you can observe while switching on or off an electric circuit having ohmic resistance only.
But do you know that if there is inductor or capacitor in a circuit, the current will not become maximum from zero or from maximum to zero in no time. The current will take a finite time or one can say it is a gradual process. Thus the electric current which takes finite time to become maximum from zero or zero from maximum in circuits having inductors and capacitors (condensers), then such time of currents are called transient currents.