Construction & chemical action of Cell:

In cells, an electrolyte separates two charge collecting materials called electrodes, to which external connections are made. The electrolyte pushes electrons onto one of the plates and takes them off the other. This action results in an excess of electrons, or a negative charge, on one plate and a loss of electrons, or a positive charge, on the other plate.
Electrolytes are chemical solutions manufactured to allow the generation and free movement of both types of ions, and are normally acid or alkaline pastes or liquids.

The action of the electrolyte in carrying electrons from one plate to the other is actually a chemical reaction between the electrolyte and the two plates. This action changes chemical energy into electrical charges on the cell plates and terminals.

With nothing connected to the cell terminals, the electrons would be pushed onto the negative plate until there was no more room. At the same time the electrolyte would take electrons from the positive plate to make up for those it had pushed onto the negative plate. Both plates would then be fully charged and the movement of electrons would cease.
If a wire were connected between the negative and positive terminals of the cell, electrons on the negative terminal would leave the terminal and travel through the wire to the positive terminal. The electrolyte would carry more electrons across from the positive plate to the negative plate. Whilst the electrolyte is carrying electrons you would see the negative plate being used up and you would see bubbles of gas at the positive plate.