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When we talk about electrolytes in biology we usually mean Sodium, Potassium, Calcium and Chloride. When dissolved in water these particles carry a charge (they are ions).
Our cells are encapsulated by membranes that are not permeable to these ions, those membranes however also contain channels that do allow these ions to pass. These channels can be opened or closed and are often coupled with a receptor, which will open/close it after receiving a correct signal.
Cells will use energy to maintain a different concentration of a certain ion inside the cell than outside the cell - the cell is polarized. This difference in concentrations is often referred to as membrane potential.
When a correct signal arrives and the channels open up the ions will move from the direction of higher concentration to lower concentration. This will cause the cell to depolarize and will rapidly change the membrane potential of a cell. This rapid change in membrane potential is called action potential(AP) and it will propagate some way along the cell membrane.
We also have channels that are not regulated by receptors, but are directly opened by AP. This means when an AP is triggered and starts propagating along the membrane, it will open new channels which again causes AP, which continues propagating, opening more channels... This is how AP can travel quickly along some very long cells (such as neurons or muscle cells) and this is basically what we mean when we talk about electricity in our bodies.