In compact matter, electrons "trickle" from a source to one atom, forcing off electrons which move to the next atom, ionizing each in turn. The same holds true in the opposite direction, where electrons are removed and the corresponding "hole", or positive ion, is propagated towards the negative electrode. Individual electrons and ions move slowly at the drift velocity, but electromotive force propagates much faster. This is similar to the motion of the particles in the toy, "Newton's Cradle".
In an ionized gas, electrons and ions may move much faster, actually crossing the full volume of the plasma (though electrons move faster, due to their lesser mass for the same charge).