If the battery voltage drops, less current flows, and eventually there is not enough to make the components work. The extent to which a wire resists the flow of current is called resistance, and is measured in ohms.
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.
This is called a negative earth system, and when buying an electrical accessory a radio, for example check that it is of a type suitable for your car system. Fitting a radio with the incorrect polarity will damage the set, but most car radios have an external switch for setting the polarity to suit that of the car. Switch to the correct setting before fitting.
The sudden surge of high current in a short circuit makes the fuse wire melt, or blow, breaking the circuit. When this happens, see if there is a short circuit or a disconnection, then install a new fuse of the correct amperage rating (See Checking and replacing fuses).