Discrete charges are a fertile representation. They are negative electrons or positive protons where humans use to live, with protons packed in atomic nuclei that make charged ions when the surrounding electrons are more or fewer than the protons.
In most cases, you don't need discrete charges, and can imagine the electric current as a liquid flow for instance. The elementary charge q is small enough that you generally don't notice it. Though, I find discrete charges easy to imagine.
Electrolytes conduct current while vacuum normally doesn't because electrolytes contain ions, even before any voltage is applied. Some molecules can easily lose one or several electron(s), others catch one or several additional electron(s) very strongly. Such molecules, helped by a solvent if necessary, become ions when one catches electron(s) from an other. The ion with more electrons than protons is negative, with one with fewer is positive.
So while the electrolyte is globally neutral (or extremely close to neutral), it contains positive and negative ions. An electric field moves these ions by acting on the protons and electrons whose numbers differ. When the charged ions move, current flows.
Over a very limited time, a "polarization" current can also flow, which is not the "displacement" current of the moving charges. Its density per surface unit is ε0×dE/dt (put signs as you want). That is, as long as the electric field E varies, a current flows through vacuum. Because ε0 is only 8.85pF/m, this current is important only if E varies very quickly. In electrochemistry it's generally negligible, especially with DC voltages and currents, but for the fast radiowaves it's paramount. This current creates a magnetic field like the displacement current does.
ε0εr×S×dE/dt, where εr≠1 in an insulator, is also called polarization current because the insulator conducts no DC current. Though, the extra current from εr>1 results from charge movements over a small distance, like ions moving from one position to the other, or electronic bonds changing their shape. That's why I mention only ε0×dE/dt as a strict polarization current.