This membrane is made up of fatty molecules that are impervious to ions. In order for cells to live they must allow material and ions to pass in and out of the cell through the membrane. They do this via special molecules that they construct called membrane proteins which sit within the membrane and allow small molecules to pass through them. There are both pumps and simple channels. Pumps use up energy and pump ions through the membrane in a single direction.
Channels simply allow ions to pass through the membrane in either direction.
Both pumps and channels are opened and closed by chemical and electrical signals from inside and outside the cell. Since pumps pump either in or out of the cell, they are probably not important for conducting ions through the cells. However, electric fish use pumps that on one side of the cell pump out and on the other side of the cell pump in in order to produce up to 650 volts at 1 ampere which can stun and kill other fish. Channels on the other hand can allow ions to travel both into and out of the cell and thus can act as conduits for ions passing through the cells. So you can essentially think of human tissues as electrical resistors whose resistance is directly related to the number and types of membrane ion channels that are open.