The kind of electricity that we channel in wires is made up of vast quantities of electrons moving through the wire. Electrons can be made to move by a magnetic field, as in a generator, or by a chemical reaction, as in a battery. Either way, the moving electrons can be sent for long distances, and can do useful work along the way. Electricity is extremely convenient and powerful, a foundation of modern technology.
With only a few exceptions like electric eels, biological organisms do not use this kind of electricity. You may know that our nerve cells use electricity. But instead of moving electrons, biology uses ions—the "charged" atoms that remain when one or more electrons are removed. Ions can move from place to place, and can do work just like electrons. Bacteria use ions to power their flagella "tails." Ions moving suddenly through a nerve cell membrane cause a change that allows more ions, further along the cell, to be able to move, creating a domino effect that ripples from one end of the cell to the other.
Ions are convenient for cells to handle. An ion is much larger than an electron, and is therefore easier to contain. But ions have to move slowly, bumping through the water they are dissolved in. Over long distances, electrons in a wire can deliver energy far more rapidly than ions in a liquid. But wires require insulation.
It is perhaps not surprising that biology hasn't used electron currents. At cellular scales, ions diffuse fast enough to do the job. And the same membranes that keep chemicals properly in (or out of) the cell can also keep ions contained where they can do useful work. But if we actually had "nerves of steel", we could react far more quickly than we do.
To use electron currents, all that's needed is a good conductor and a good insulator. Carbon nanotubes can be both conductors and insulators, depending on how they are constructed. Many organic molecules are insulating, and some are conductive. There is a lot of potential...