Fuses (or circuit breakers) are protective devices that will blow to shut-off a circuit with a fault. That purpose is usually quite obvious. But most people believe they protect the device, and that's usually wrong. If a device has developed a fault that blows the fuse, the device is already faulty. Mostly they protect against a continuing and/or worsening fault from causing a fire. The device itself, the wiring that leads to the device, and wiring in the wall.
With a stereo system, it might be plugged into a wall socket with a circuit breaker, there will be a fuse or breaker in the house's fusebox, there'll be a fuse inside the amplifier's power supply, there'll be a fuse between amplifier and speakers. If you accidentally short out the speaker wire, you'll probably blow the speaker fuse, it's highly unlikely that an amplifier fault will blow the speaker fuse and not wreck the speaker (the voltages normally used to drive the speaker are too similar to the high voltage supplies in the amplifier output stage, so the fuse must be able to withstand that, and wouldn't blow quick enough to protect the speaker against a prolonged fault). If the amplifier developed a serious fault it may blow the fuse in its own power supply, though it may just burn out some components. If the amplifier's power transformer developed a short, it'll probably blow a fuse in the amplifier, or even in the transformer. If the amplifier's power cord got damaged, it may trip a breaker in the wall socket or fusebox.
Fuse ratings
Fuses are rated for the current they can carry without blowing, and the voltage that they can break, at an average room temperature. For example, a 5 amp 250 volt fuse will pass 5 amps of current without breaking, and can break up to 250 volts. In high temperatures they'll blow more easily, and in cold environments will pass more current before blowing.
In general, fuses are installed according to the manufacturer's recommendations for the wiring or equipment. You shouldn't vary from the recommendation without good reason (e.g. the manufacturer got it wrong, which would be fairly rare but not impossible).
Voltages higher than the fuse rating may arc-over inside the blown fuse, and continue to conduct. The correct type of fuse should be used for the situation. You may have noticed that 250 volts is higher than your normal mains voltage, that's because the normal mains voltage can vary and the fuse needs to be suitable for the highest voltage that may occur. 250 volts can be within the allowed range in your country, so it shouldn't nuisance trip, but it will still do its job when breaking any voltage lower than that rating.
As to how much higher than 5 amps before it breaks, that will depend on the fuse manufacturer's specifications.
There are slow-blow, fast-blow fuses, and very-fast blowing fuses.
The general purpose thin strip of wire in an ordinary fuse is a fast-blowing, or rapid, fuse. It's probably the most common type of fuse used inside equipment. If the equipment develops a fault it should blow its fuse and protect the supply wiring from catching fire. It's rare that an internal fuse will blow to protect the equipment from damage, other than from further damage, the fuse has normally blown because the equipment went faulty. And fast-blowing fuses will be the only type of fuse in (vintage) household fuseboxes.
Slow-blow, or time-delayed, fuses are used for devices which may use more current at switch on than during normal operation (such as internal fuses in some stereo amplifiers). Using a fuse type more tailored to those conditions allows selecting a fuse closer to the actual current usage for better protection, instead of using a much higher rated fuse to avoid nuisance tripping. They often look like a fuse wire wound around a tiny core.
Very-fast blowing fuses are used when its desirable for the fuse to blow as fast as possible. They may look like a thin wire with a spring attached at one end under tension (to break the fuse apart faster when it starts to melt). They may be a ceramic fuse filled with sand (to quench any arc quicker than a fuse that just has air between the end caps). Stage lighting usually used ceramic fuses between dimmers and luminaires in an effort to better protect the dimmer circuitry from damage should a lamp blow and go short-circuit (which can happen if the base of the lamp is below the filament, a blown filament can break apart and drop shorter sections of itself down onto the lamp's internal conductors drawing very high currents).
Choosing fuse values
In most cases, your only choice is to replace like-for-like. If your 16 amp fuse blows in the fuseboard for the power points, you choose 16 amp fusewire from your spares to replace it with. Likewise for other fuses, you use the value specified for the equipment.
About the only time you can choose a fuse value is when you're designing equipment. You'd measure your device's normal current flow, and measure any switch-on in-rush current above the normal usage, and choose fuse values to suit.
Nuisance tripping
Generally speaking, fuses don't blow unless there's a fault, and that fault should be diagnosed. With fuses for general purpose power outlets around your house, sometimes that's a damaged cord or badly fitted plug, but often the only time they blow is when too many things are plugged into the same circuit.
There are some times that a few will die without their being a fault. This is usually only on high-current circuits, like water heaters, where after many years of a heavy load going through the fuse, it has weakened. This is a rare occurance.
If a fuse blows again immediately after replacing it, that's not a nuisance trip. You have a fault, and it needs finding and clearing.
Household fuses
With household wiring, protecting against wiring catching fire is definitely the prime concern. The choice of fuse size is determined by the electrical wiring in the walls between the fusebox and the powerpoints, and similarly for lighting circuits.
The wiring has a safe current limit that mustn't be exceeded (whether by faulty equipment, or by connecting too many things that draw too much current), the fuse size is chosen according to the specifications of the wire. If the fuse blows, you have probably overloaded the circuit, and the solution is to remedy the fault, not to put in a bigger replacement fuse (unless a too small one had been fitted).
This would mean either unplugging some of the too-many things you have connected, disconnecting something that uses more current than the circuit is able to provide, or disconnecting something that has a fault.
If the problem is down to you plugging too many things in at once, but you need to use all that equipment at the same time, you either have to move some of it onto a different wiring branch, or have another wiring branch installed by an electrician.
Lighting circuits usually have lower fuses and thinner wiring than power point circuits, since they're not expected to have high current devices on them. Though some countries don't do that, their lighting circuit wiring is just the same as general purpose outlet wiring.
Very high current circuits, such as water heaters, have special considerations. Either heavier guage wiring would be needed than general purpose wiring, or the wiring mustn't be cemented into the walls (allowing a bit more air-cooling), and each device uses a dedicated wiring circuit.
Equipment (internal) fuses
Electronic equipment may have several fuses in it. But going back to the household wiring; firstly, the house fuses will protect the wiring in the walls should your equipment (e.g. a tv set) have a severe fault that draws extremely excessive current, or if it's power lead to the wall socket gets sliced into. Inside your equipment will be a fuse somewhere between its power cord and its internal power supply, this will blow if the device develops an internal fault within its power supply that draws excessive current that could be a fire risk. There may be fuses between its power supply and other parts of the device which will blow if some other part of the device develops a fault that draws excessive current that risks setting its power supply on fire. It's a daisy-chain of protective fuses. None of which will probably save your tv set from having a serious breakdown that requires repairs, but at least it shouldn't burn your house down.
Circuit breakers
These do essentially the same job as a fuse, tripping with excessive current and breaking the circuit. Often with some delay. A certain amount of overcurrent can be drawn for quite some time before it trips, a larger overcurrent draw will trip it quicker. An extremely excessive amount will trip it instantly.
It's just a device that can be easily reset without having to be rewired. That can be a problem in itself, as people just reset the breaker without resolving the fault condition that tripped it. But, at least, fixed-in-place breakers stops people putting their fingers into electrical terminals, or rewiring fuses with the wrong gauge fusewire. Unpluggable circuit breakers don't prevent that, though.
Earth leakage detection / ground fault detectors
These detect when some of the current is leaking out of the circuit, and is going through something else (possibly you) instead of where it's supposed to go.
In your house, the electricity comes in from the street, goes through your fuseboard, goes out to the powerpoint, goes into your equipment, comes back out of the equipment, back through the power point, and back to the fuseboard. It makes a circuit, hence why they're called electrical circuits. The amount of current going out has to be exactly the same as what comes back.
In a fault where some of that current goes elsewhere (where it shouldn't go), there will be an unequal amount of current going out of the fuseboard to what comes back to it. The protective device will detect this imbalance and switch off the power. It's very sensitive, it has to detect an imbalance of about 30 mA difference (0.03 amps), the tiny amount of current that could kill you, even though there could be 10 amps of current actually going through the circuit.
Unfortunately they are prone to false triggering. Virtually every electronic device leaks a very tiny, insignificant, amount of current through its insulation and electronic components, even when they're working perfectly fine. But when you have a modern household full of electronic devices all leaking that tiny amount, they all add up to enough to upset the leakage current detection device.
The solution to that problem is instead of having one earth leakage detector protecting everything, is having individual detectors on each branch circuit. That way the individual detectors only have to cope with a smaller number of electronic devices, which shouldn't take their detector over the threshold when there isn't actually a fault.
This is about the only type of fault detector that can protect you from being electrocuted. The amount of current it takes to kill a person is tiny compared to the amount of current it takes to run electric devices. It's not foolproof though, it only detects a current imbalance within a circuit. If you happen to get a shock where the current goes through you and back into the same circuit, it can't detect that as a fault.
The moral of that story is keep your hands outside of electric equipment, and get things competently repaired when they get damaged. Also keep liquids away from electrical equipment, it can conduct electricity through all the things that normally keep it isolated from you (i.e. its casing).
Surge arrestors
Where I live the mains electricity is 240 volts, nominally. It's allowed to vary about 10 volts from that and not be considered a fault. Also, all day long, there can be incredibly brief moments where spikes of several hundred volts are on top of that 240 volts. This happens as things are switched on and off elsewhere on the electricity grid (equipment in industry, the switching yard at power substations, every household in your area, etc).
Mains power has always been like that, and always will be. Everything that is connected to the mains has to be able to cope with those spikes. That doesn't mean your stereo system isn't allowed to make audible ticks as your fridge compressor switches on or off, but it does mean that your stereo should not break down from it. If it does, it was never fit for purpose to be plugged into the wall. Likewise for your tv or computer. Equipment plugged into the wall must be able to tolerate the normal working conditions of the mains electricity supply, and brief spikes are part of that.
Unfortunately, snakeoil salesman will tell you that you need to buy a surge arrestor to protect your expensive tv set from mains surges, and perhaps your shonky insurance salesman, too. Even some tv manufacturers tried to tell you that your warranty would be invalid if you didn't use one (when really they should have built their tv set better). Once more, I'll say read the previous paragraph about spikes on the mains wiring (they're normal, continuously occuring, and anything that's plugged into the wall has to be fit to be plugged into the wall without requiring extra protection).
Also, I'll point out that the surge arrestor really can't help with that problem. Those spikes are too short, and too small (relatively speaking) for a surge arrestor to fully suppress, if at all. If you tried to suppress all spikes to that extreme, it'd be working hard all the time, getting hot, blowing fuses, and having a short life.
Probably the biggest instigator of stores selling overpriced crappy surge arrestors to the general public was plasma tv sets. In their early days, a few went up in flames, and some of their highly flamable chemicals in the screen were extremely hard to extinguish. Most other electronic devices will just go phhftt and die when zapped.
What a surge arrestor could actually do is deal with something like when a tree branch falls on mains power lines, or a lightning strike hits the street wiring, and puts a huge amount of current where it's not supposed to go. A surge arrestor should clamp that excessive surge down by a significant degree, and blow their fuse while doing so, perhaps blowing the fuse in your fusebox, preventing your electronic devices from catching fire and burning your house down. It probably won't protect your device from being damaged, though.
And the best place for your surge arrestor to be is installed in your fusebox by an electrician, not a powerboard plugged into a wall socket with your tv set plugged into that. Firstly, it'll protect the whole house, and secondly, it'll be a much better designed suppressor. In some countries, they're a part of the electrical installation of the house, already.
And, of course, the electricity companies have arrestors along parts of the street wiring and in their switching yards. You could live to be 80 and never been aware of any mains surges, because the electricity company has been dealing with them year after year.
EMI and noise filters
EMI (electro-magnetic interfence), or noise, on the mains wiring is a daily occurance (it's normal). It can cause clicks, buzzes, whines, to be heard on your stereo system, tv, or other devices with speakers. And although a really good stereo amplifier could be designed to filter that noise out to the point you wouldn't notice it, few do.
You can buy power boards with noise filters built into them, and they're about the only “fix it” gadget I would bother with, though they're only partially effective. (Powerboards with other fancy features are generally providing useless features.) But a less annoying POP noise, when your fridge turns on and off while you're listening to music is an improvement, even if you can't completely eliminate them.
For dealing with that kind of nuisance electrical noises, putting a mains filter on the device that's affected by the noise (your tv, your stereo, etc), and another filter on devices that create noise (fridges, air-conditions, pool filter motors, etc), may help make things quieter.
But if you have something like a fridge compressor that makes your stereo emit a very loud bang, it might be worth having it serviced. It may have a motor or thermostat switch fault. Normally, you should barely be able to notice that kind of thing.
Our mid-1960s Australian house has all the powerpoints in the entire house on one 16 amp fuse. All the power points are on a single wiring run that snakes around the perimeter of the entire house. If we were to try and use two 2400 watt electric heaters in two rooms at the same time, we'd blow the fuse (and this is a three-bedroom house). Unless we re-wired the house (something that's highly impractical because all the walls are brick, there's virtually no underfloor access, and very limited roof access, never mind the enormous expense), the only solution is to use room heaters at a lesser wattage. A similar problem exists if you had an electric heater on in one room and plugged in an electric kettle in the kitchen.
Unfortunately we're stuck with the problem of a poorly designed house, with no central heating, no insulation, and wiring scheme that didn't think beyond people using 500 watt bar radiators to heat the bedrooms, and a non-electric kettle on the gas stove. That, or they expected you to rug-up and scrape-by in damp cold rooms, which is bad for your health. There is a gas fire in the lounge room, but that can barely warm the whole house if you leave all the internal doors wide open. And if someone switched on the exhaust fan in the bathroom, the whole house goes chilly within minutes. I have nothing but cold disdain for cretins who design houses with such utter contempt for their owners.
It's almost like electricity didn't exist, here, only the kitchen and the lounge room have more than one power point in them. They have two each, oh the luxury! The kitchen has one behind the fridge, and one behind the gas stove. We revelled in modern conveniences in the 1970s and got an electric toaster, which means the gas oven had to be shoved over away from the counter and towards the door to uncover the power point. The lounge has one powerpoint either side of the gas fire. And all the power points throughout the house are singles, so that meant double double-adaptors were used all over the place; and extension cords, since few of the power points are actually in useful places.