Step 1: Why a new current was needed.
Consider charging a parallel plate capacitor. Apply Ampere's law \(\oint \vec{B}\cdot d\vec{l} = \mu_0 I\) using two surfaces bounded by the same loop: one surface cut by the wire (gives current \(I\)) and one bulging surface passing between the plates (gives current \(0\)). The two answers disagree, so the law is incomplete.
Step 2: Maxwell's fix.
Between the plates the charge \(Q = \varepsilon_0 \Phi_E\), so a rising charge means a rising electric flux. Differentiating, \(\dfrac{dQ}{dt} = \varepsilon_0 \dfrac{d\Phi_E}{dt}\). The right side is defined as the displacement current \(I_d\).
Step 3: Definition.
Displacement current is the current equivalent of a changing electric flux:
\[I_d = \varepsilon_0 \frac{d\Phi_E}{dt}\]
It restores continuity, since \(I_d\) between the plates exactly equals the conduction current \(I_c\) in the wire.
Step 4: Contrast with conduction current.
Conduction current \(I_c\) needs a material with mobile charges and obeys \(I_c = nAev_d\). Displacement current needs only a varying field and flows through empty space. One is transport of charge, the other is a field effect; both generate identical magnetic fields, which is why light and radio waves can travel through vacuum.
\[\boxed{I_d = \varepsilon_0 \dfrac{d\Phi_E}{dt}}\]