[Coda]

Energy OF electrons, not energy FROM electrons. But otherwise, close enough.

You are 100% correct that electrons are moving. Electricity on its own, however, DOES require contact, at least at the macro scale. That's what wires are for -- the electrons in the metal atoms are able to move without a lot of resistance (sort of analogous to friction), so when you pull electrons out of one side of a wire, electrons will ripple through to fill up the void you created. If this were to leave a void on the other side, though, the electrons would just flow back. (Yes, this means that electrons are constantly swirling around in metal, but if you think about it, you already knew that electrons are constantly swirling around -- it just means they're able to swirl around in a little bit bigger of a range than you might have expected.) In order to get an actual electrical current, then, you need to have a source of electrons on the other end to refill what you took out.

But electricity isn't the only effect in play here.

When electrons move, they create a magnetic field. And when magnetic fields move, they cause electrons to move. This is how windmills generate electricity, actually: the rotation moves a bunch of magnets along a bunch of coils of wire, so the magnets sort of drag electrons along, pushing some electrons down the wire and pulling some up from the other end. If you put a load between the far ends of the wire, like a light bulb, then you can make those moving electrons do work.

There are also electric fields. Protons have a charge of +1. Electrons have a charge of -1. A normal atom has equal numbers of protons and neutrons, so the net charge of that atom is 0, and so the net electric field of that atom is zero. But if an atom has more or fewer electrons, then it gets a net charge. We call such a charged particle an "ion" and it has a negative or positive charge, and it has an electric field with a strength and polarity based on that charge. (Ions can also be formed by multiple atoms bonded together; bonded atoms "share" electrons in a sense, so if any of those atoms are missing electrons or have excess electrons, the whole bonded group acts as an ion.) Oppositely-charged ions attract each other; ions with the same charge repel each other.

In clouds, you have ice crystals banging around a lot, and sometimes when they do, an electron will move over from one to another. This gives one crystal a positive charge, and one crystal a negative charge, and usually (for reasons that aren't well-understood) the bigger crystal is the one that ends up with a negative charge and the smaller one ends up with a positive charge. Those crystals do attract each other, but the attraction is fairly weak, and they have so much kinetic energy from being blown around in the air that they get separated. The heavier negative crystals filter towards the bottom of the cloud and the lighter positive ones filter towards the top. This creates a charge differential -- that is, an electrical potential, a place where an excess of electrons desperately wants to move over to a place with a shortage of electrons. Cloud lightning happens when the potential gets so great that it overcomes the resistance of the air (just because it's very high doesn't mean it's infinite) and the electrons take the shortest path from negative to positive.

Lightning strikes happen because that massive negative buildup in the cloud attracts positive charge in the earth to amass at the surface below. It takes a LOT of potential to overcome THAT much air, but as soon as even two little points get enough charge to get past the resistance of the shortest route between them, BOOM.

As for a potato battery: No, one piece of metal isn't enough. It requires two pieces of different kinds of metal (usually copper and zinc). There's phosphoric acid in the potato, and the speed that the acid can cause chemical reactions is entirely dependent on how fast the it can ionize the metal atoms, which is why you need two DIFFERENT kinds of metals -- one kind of metal will part with its electrons more readily than the other, and so that chemical reaction can happen more easily. This creates a charge differential, and if you connect a wire to the two kinds of metals, electrons can flow through it from the slower side to the faster side, facilitating that reaction.

As for why it doesn't pull the electrons back, the zinc ions react with the acid, creating zinc phosphate and positively-charged hydrogen atoms, which are drawn towards the now-negatively-charged copper. They pick up those extra electrons to become hydrogen atoms (and then react with other hydrogen atoms to create molecular hydrogen gas, which you can see bubbling at the copper metal if you set it up in a beaker instead of inside a potato). And so Newton's laws are conserved and the net charge of the system as a whole remains balanced, but you've derived work from the energy stored in chemical bonds in the potato.

Eventually, the potato will run out of phosphoric acid around the zinc metal, because it'll have all turned into zinc phosphate... and so the battery will be dead -- no more chemical reaction means no more ions means no more electron flow.

Some chemical reactions, such as the ones used in lithium batteries, create compounds that can be broken back apart easily by shoving a little bit of energy at them. And then if there's no path for electrons to flow (that is, you disconnect the load), then that reaction will sit there in tension, waiting for a source of electrons to let it finish.

Solar panels basically work by having a special kind of metal surface, so when a photon hits it, it knocks an electron off, so an electron gets pulled in to fill the void, and that creates a current flow.

Coal is a completely different thing. You can't actually create ELECTRICAL energy from coal. Instead, you burn coal, adding a little bit of energy to break apart a big molecule, and breaking the molecule apart releases more energy than it took to break it, so that creates heat. The heat is used to boil water, and the steam is used to spin a turbine, and the turbine is used just like the windmill above. Why yes, burning stuff IS an inefficient way of using energy.

Nuclear energy works the same way, except instead of breaking apart big molecules by setting them on fire, you have big atoms that flake off pieces of themselves on their own (radioactivity), and when they do that the particles they release have a chance of hitting other big atoms and making them split (atomic fission). In both cases, that produces energy... which gets used to boil water to spin a turbine to spin a magnet through a coil of wire. Nuclear power plants are actually not especially efficient either, but nuclear reactions generate SO MUCH raw energy that it doesn't much MATTER.

EDIT: O.o 55 aurum megapost

EDIT 2: I was wrong about the specific details of the chemistry of a potato battery. I've corrected that.