Someone earlier asked me something about electricity and how it works, and we all know how little motivation I need to babble effusively about physics. So here we have it, a primer for lay people on how electricity works.
The basic principle that you have to be familiar with to understand electricity is charge. there are two types of charge, called positive and negative1. If two objects have the same charge (both positive or both negative) then they repel each other, whereas if they have different charges (one positive, one negative) they attract each other. The bigger the charges, the bigger the attraction or repulsion will be. This much, you'll just have to take on trust. It's one of those things that "just is"2. From here though, mostly we'll be working out how one thing stems from another, and won't have to take things on trust.
If you have a metal wire it contains lots and lots of tiny particles called electrons, each of which has a tiny little negative charge. If you put something else that's charged at one end of the wire, the electrons will then be attracted to/repelled from it, and move down the wire. As they move down the wire they can hit bits of whatever they're passing through, and give up a little bit of their energy. That's how electricity powers things.
Also, let's say that we put a negative charge at one end of the wire (A), and a positive charge at the other end (B). Our little electrons will be repelled from A and attracted to B. This would have two other effects though. Firstly, since the electrons had left from A, there wouldn't be any electrons left there3 to be able to go over to B. Secondly, lots of negatively charge electrons would be leaving A, so it would lose its negative charge, and lots of negatively charged electrons would be arriving at B, which would counteract its positive charge. With nothing to power it, and no electrons available to flow, a whole lot of absolutely nothing at all would happen.
This is why, in order for electrical things to work, you need to things. Firstly, you must have a complete circuit. this ensures that the electrons going out of one end are then effectively fed back in at the other end so you never run out. Secondly, you need to have a power source like a battery or a mains socket, which will stop the charges from balancing out to zero.
OK, so that's the basics. So what are volts and amps and watts and all those other things. Amps (short for amperes) are the unit of current, which measure how much charge is flowing per unit time. One amp is one coulomb per second. So if you have a current of 5 amp, that means 5 coulombs of charge are passing through each point of the wire every second.
Now, imagine that you're told that you have a road and that cars are travelling down the road at a rate of 10 cars per second. This could be a narrow road with only one lane, with lots of cars going very vary fast, or it could be a huge ten-lane highway, with the same number of cars spread out over ten lanes and each one going at one tenth of the speed as the cars on the narrow road. Either way you'd get ten cars per second. Electricity is much the same; you can have a 5 amp current by having densely packed electrons moving very fast, or by having a thicker wire and having them move ore slowly.
Every wire has a resistance (measured in ohms) which is analogous to how wide the road is. This is partly down to the actual thickness of the wire, but also depends on what the wire is made of. For good conductors like copper or gold, the "lanes" are narrower, so you can fit more lanes in the same area.
So, let's say I know that a wire has a certain resistance, and I want to have a certain current go through it. How do I do that? Let's go back to the road analogy. If I wanted to make the cars fall down from one end of the road to the other, I might put the road on an incline and make them roll down the hill. If I had a high resistance (narrow road), I'd have to have a steeper incline to create the same current (number of cars per second) as I would with a lower resistance wire (wider road).
The equivalent of this idea is voltage, which is measure in volts. Voltage is also called potential difference, because it's the difference between the two points that matter. The absolute height of the two ends of the road don't matter; it's the difference in height that's important. I you want to make a physicist twitch, mention a "voltage through something". This doesn't make any sense! If you want to make a physicist love you forever, refer to the voltage across something which is right.
A volt is one joule per coulomb, with a joule being a unit of energy that's brought in from mechanics4. So if you have a 9 volt battery, this means that for every coulomb of charge that's flowing it gets given an energy of 9 joules. The more energy an electron has, the faster it can go, so if you have the same wire, a bigger voltage across it will induce a bigger current than a smaller voltage.
So what about watts? A watt is a unit of power, which is physicist speak for "energy over time". In fact, a watt is one joule per second. A 60 watt light bulb uses 60 joules of energy every second. This energy comes from the movement of the electrons down the wire; as they ass through the filament, they bang into the other particles of the wire, making it heat up and glow, and then they come out the other side moving more slowly than they went in. Now, remember that an amp is how much charge is flowing every second, and a volt is how much energy each bit of charge has. This means that the power (watts) is equal to the current (amps) multiplied by the voltage (volts). So let's say we're in Britain where mains electricity is 230V and we're using a 60W light bulb. this means that a current of about a quarter of an amp is passing through our light bulb.
You may have heard the expression that it's amps that kill people and not volts. This isn't actually true. It's actually a combination of the two. You can't have a current (amps) without a voltage. It just doesn't work. It's like expecting a car to spontaneously start rolling down a perfectly flat road. On the other hand, you can have a voltage without a current – that's the equivalent of a steep road but without any cars on it – which is the reason for the expression.
There's a lot more that I could cover. The difference between alternating and direct current (AC and DC), capacitance, inductance, electric generators, series and parallel circuits, and so on and so forth, but I suspect that anyone who's managed to get all the way through this has their eyes glazing over by now, so I'll stop.
Any questions? Any bits that aren't clear? Any other points you'd like me to explain?
 The + in my userpic on this entry is because the rho meson that's depicted in it has a positive charge.
 Actually, I think that it can be explained at a more fundamental level with gauge symmetries and quantum field theories, but "it just is" works well for me.
 Actually, there'd be lots of electrons still there. Most of the electrons are bound up and aren't free to flow. It's the free electrons that could potentially run out.
 In an uncharacteristic moment of willpower, I'm resisting the temptation to tangent off into a complete guide to mechanics here as well.
Just keep on trying till you run out of cake
- On electricity (warning! contains physics!)