Conduction current: the actual current due to the flow of free charge carriers (electrons) through a conductor.

Displacement current: an effective current, $i_d=\varepsilon_0\dfrac{d\Phi_E}{dt}$, that arises from a time-varying electric field/flux in a region — it does not involve actual motion of charge in that region (e.g. the gap between capacitor plates).

In a circuit with a capacitor driven by a time-varying source: conduction current $i_C$ flows in the connecting wires as the capacitor charges/discharges, while displacement current $i_d$ 'flows' in the gap between the plates due to the changing electric field there.

Maxwell showed that at every cross-section of the complete circuit (including the capacitor gap), the total current is continuous: at any instant, $i_C$ in the wire equals $i_d$ in the gap, so the sum of conduction and displacement current gives one continuous current throughout the circuit — resolving the apparent discontinuity of conduction current at the capacitor.