How does a simple component and continuity tester work?

A continuity tester is sometimes referred to as a "bell set". To make one, procure a battery, an electric bell or buzzer, and some wire. With wire 1, connect one battery terminal to one side of the bell. With wire 2, momentarily connect the other battery terminal to the other side of the bell, and note the buzzer sound. Any time you complete the circuit, you will hear that sound.

Now create these connections: battery+ => wire1 => bell => wire2 => component => wire3 => battery-

If the unknown "component" conducts current, then you will hear the bell, otherwise there is a fault.