Most are wired through the ignition switch, so that they work only when the ignition is switched on. This prevents you accidentally leaving something switched on which might cause the battery to go flat.
Depending on the fuse design, it is sometimes possible to tell whether it has blown by holding it up against a light; a break in the wire inside may be visible. Another clue is blackening of the glass cover. If there is no visible sign, check by fitting another fuse of the same rating; if that cures the trouble, then the fuse was to blame.
If the cable overheated, there could be a fire. The fuse prevents that happening, because its thin wire will melt and break the circuit long before the cable itself can heat up and burn. Some cars have only two fuses. One rated at about 30-50 amps protects components wired through the ignition switch — flashers, wipers, heater-motor and instruments.
Some components must be connected in series. For example, the sender in the fuel tank varies its resistance according to the amount of fuel in the tank, and sends a small electrical current to the fuel gauge.