Some accounts of this are pretty misleading, and I'll try to tell it straight though I might fail at that.
First off, electricity is electrons moving. If you have a DC current with a battery, 1 ampere means there are 6 quintillion electrons per second leaving one battery terminal and 6 quintillion electrons per second entering another battery terminal, against resistance. That's what does the work.
Now here are two stories about how it happens. I think the first story is less true, but I can't say it's completely false. Imagine there were no electric fields at all, but electrons could travel freely through copper wire. When you have an electric circuit with a battery in it, there are a whole lot of electrons at one terminal, and electrons that enter the other terminal get involved in a chemical reaction so they don't get out again. So electrons would diffuse through the metal. More of them leave one terminal than come back, simply because there are more of them there. More of them enter the other terminal than come back, because there are fewer available to leave. Diffusion alone would be enough to give us an electric current. As long as electrons travel very fast in random directions, on average they will travel more from high concentrations to low concentrations. That would be enough.
However, we know that there are electric and magnetic forces.
Now I will explain magnetism. I warn you that some people believe my explanation is wrong. So check it for yourself, or be cautious. For historical reasons, we knew there were magnets, and we thought magnetism was something fundamental. But if you look carefully at the equations, there is no magnetic force unless the source charge is moving. And magnetism has no effect on a target charge unless the target charge is also moving. Magnetic force depends entirely on the frame you choose as an observer. It is a fudge factor, because the electric field equations do not otherwise account for frames and are only correct in some frames.
If you're with me that far, then it turns out that electrons that move at relativistic speeds in wires, have some consequences. An electron's electric field is stronger in front of it than behind it. So it is attracted MORE by the positive charges ahead of it than by the positive charges behind it. But it is not affected more by negative charges ahead of it, which are traveling at the same velocity. The very fact that there are electrons traveling through a wire, creates a force to speed them up. So when a circuit is first formed, only a little bit of current travels through it, and the amount increases until it reaches an equilibrium with the things that tend to slow it. And if the battery stops providing voltage, the fast-moving electrons don't stop instantly because their own velocity gives them a decaying force to keep them going awhile. This is called "hysteresis".
There are various other effects of relativistic electron velocity, that explain various properties of electricity.
Well, but there's the claim that electrons in fact don't travel at close to lightspeed. The fields travel that fast but the electrons don't. You can calculate the average speed of electrons in copper wire, and it's very very slow. This might be misleading, though. First off, copper has 29 electrons and only one of them can move. And then maybe the ones that can move are in fact moving at relativistic speeds, but mostly in random directions. When you get an electric current, a small extra fraction of them travel in the direction of the current. Not very many, only a quintillion or so per ampere, but that's enough to give you the observed current. All the rest of the motion averages out. So on average they move very slow, but the ones that make the current could be going fast. They're just a small fraction of the whole.
Well, but I've seen the claim that the average random movement of electrons apart from electric current is fast but not nearly that fast. Maybe I'm wrong.
Maybe all the simple answers are too simple, and the complicated answers are not easy to follow.