Atoms are largely made up of protons and neutrons, with electrons whizzing around a central core. If some of those electrons are stripped off and sent down a wire, THAT is electricity. Most of our domestic electricity comes from driving magnets around inside copper coils, using water or steam power - this generates a reliable steady stream of electrons to fill our power grids with what we call electricity. Yes - lightning is an extreme form of naturally occurring electricity, but it is too powerful to do anything useful with.
If I remember chemistry correctly, charge is the excess or deficiency of electrons in an object. Electricity is caused by (or simply is) the movement of these electrons between two areas or objects with unbalanced charges.