Some ancillary components can be operated without the ignition turned on by turning the switch to the auxiliary position. A radio is usually wired through this switch, so that it can be played with the engine off.
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.
Most car handbooks and service manuals include a wiring diagram which can be difficult to follow. The colour-coding, however, is a useful guide to tracing wiring.
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.