Basically the electricity flows from the negative to the positive electrical input terminal or vice versa.
What happens in between to alter it is the circuit. Major components are resistors that limit current, capacitors that store it, and transistors and valves that amplify it. Coils are used to tune frequencies, and diodes restrict flow in one direction only.
Basic circuits only needing a few of these chaps are a simple transistor radio or sound generator, which also use knobs which are the variable resistors (to alter volume) and capacitors (frequency). Integrated circuits are miniaturised versions where everything is there but in microscopic sizes, but still does the same thing.
Bulbs, leds and lcds are also often added to indicate all sorts of data besides on/off status etc.