I never really understood short circuits and blowing fuses with too many things plugged in until I was 61 years old. Yes, I knew that the problem was too much electricity going through the wires, which would then get hot enough to cause a fire, so a fuse is installed with a soft metal that will melt and break the circuit before the rest of the wires heat up too much. Why does it get too much electricity, though? It is easy to see why a fuse would blow if something external introduced too much electricity to the system--- a lightning strike, for example. But what is the problem with too many appliances plugged in? Surely that would soak up more electricity, not introduce more of it. The appliances are consumers, not producers, after all.
But that reasoning is wrong. True, the appliance are users, not producers. But they are also, when turned on, new channels through which the electricity can flow. That's they key.
Start with the idea of how much electricity there is. That is called the charge, and is the basic scientific unit for electricity (measured in "coulombs" a very ugly name). The charge is made up of lots of electrons or of lots of electrons being missing, which makes for an imbalance either way. When you get opposite charges on each end of a wire, electricity starts flowing between them to try to equalize the charge, and we have a "force". We measure the difference in charges as the "voltage", which shows how much charge is affecting the wire circuit. Charge is like water flowing through a pipe, and voltage is like the pressure that moves the water. There can be a lot of water, but it doesn't matter how much if the pressure is low: it's the pressure that matters. But it isn't only the pressure that matters. The pipe is of a limited size, and the flow of charge per second is called the "current", measured in "amperes".
This gets us close to the problem of blowing a fuse. If there's too much current, that's the problem of too much electricity overloading the wire. So now let's think of why there might be too much current. Too much charge and voltage is the obvious answer. That's why lightning could blow the fuse. But household voltage is fixed at 110 volts ( 120, even if labelled as 110), so too much voltage is not the usual problem.
But there's something else that affects the current besides the voltage: the pipe. Remember, it's of limited size. So a narrow pipe would let less water per second through than a wide pipe. The same is true of wires. There, we call the difficulty the charge has in getting through the wire easily the wire's "resistance". A thin wire has more resistance than a thick wire; it's harder for the electricity to flow through quickly in a large amount. It's not just the width, either. I don't know yet if the length matters, actually--- will a long wire have more resistance than a short wire? (It does dissipate more electricity as heat, but that's different, maybe.) But what the wire is made of definitely matters. Copper has low resistance. Plastic has high resistance. Aluminum has pretty low resistance. Silver has the lowest resistance of any metal.
When the wire starts a circuit away from the main line coming into the house, it actually doesn't make a complete circuit until you turn on a switch. It starts off from the main line, but then it comes to a break in the line, an air gap, which isn't close until the switch is pulled. But the switch sends the circuit through something that we want to turn on, a light bulb for instance. When it is pulled, the electricity continues through the wire to the light bulb. There it encounters *lots* of resistance, because it has to go through a narrow tungsten filament. There is so much resistance that a lot of the electricity turns into light and heat as the filament glows. This cuts down the current a lot from what it would be without the light bulb there. In fact, if you just connected the entire circuit without any light bulb's resistance in the way, the current would be huge--- with that voltage and almost no resistance, the current would be able to get through very easily and there would be lots of charge flow per second. Nonetheless, that much current would heat up the wire, and might even make it glow and start a fire in the wall the wire is going through. The fuse would still be in the circuit to the main line, though, and the high current would melt the fuse, introducing an air gap, and the current would stop flowing before it could burn down the walls. So having the light bulb to add resistance in the circuit is crucial.
A short circuit happens if two of the wires in the circuit get too close to each other, so the charge doesn't have to go through the light bulb any more. If they actually touch, then if the switch is off, all the charge can just go through the wire anyway, and the fuse will blow. If the switch is on, most of the charge will still go through the wires instead of the light bulb, since that's the easy path--- the path of least resistance. So the fuse will still blow. Also, it isn't actually necessary for the wires to touch. If they just get very close, the charge will jump across the air, because air is just something with high resistance and if the air gap is small enough it provides no more resistance than the light bulb. The problem then is not that the fuse blows, but that you've got sparks in your walls and they catch fire and burn down. So a short circuit is bad news either way.
Now suppose there is no short circuit, but we put a second light bulb in the circuit, one with its own switch. If we just put the light bulb switch further down the wire, that would be pretty stupid, because if only one switch was turned one and the other one was off, the circuit would be broken and neither light bulb would turn on. On the other hand, if we just had one switch, we'd have to turn both light bulbs on at the same time. Another problem is that the first light bulb would cut down the current quite a bit with its resistance, maybe not leaving enough for the second light bulb. That is called attaching the light bulbs "in series".
So, instead, we attach the light bulbs and their switches "in parallel". The wire goes along, and if the first switch is pulled, the current can get through the entire circuit. If, however, the second switch is pulled, the charge can go either through the continuous connection through the first light bulb OR through the second, but doesn't have to go through both, as it would have if they were "in series". Instead, half the current can go through each light bulb. That means, also, there is enough current for both, because the total current can be bigger, with each half going a separate way through the resistance. It doesn't all have to crowd through that first filament; part of the charge goes through the second filament instead.
We can keep on adding light bulbs. We can turn on three, and then four, and then five, and just keep going. But there's a problem. As we turn on more light bulbs, we keep providing more ways for the charge to get through the resistance. The total current can get bigger and bigger with all those different ways to get through. As it gets bigger, though, we're getting closer to the situation with just a wire and on light bulb to throttle the charge and reduce the current. Eventually, with 20 light bulbs turned on, the resistance of the circuit is so low-- being spread across lots of parallel light bulbs-- that the wires get hot, and the fuse blows. This is the classic problem of a blown fuse-- you have too many appliances turned on at once. You have too many pathways for the charge to get through the circuit, and the current gets too big.