How a Typical Bosch Relay Works
This is a typical Bosch relay. Ford's version is typically a light gray and no mounting ear while Chevrolet units are usually black. Most aftermarket relays will have a mounting tab and should always be mounted with the contacts facing down.
The innards of the relay. When power is applied across terminals 85 and 86, current flows through a coil of small wire. This wire is about 100' long and is usually 28 gauge. This builds up a magnetic field in the bar it's wrapped around, and the steel plate snaps to it. When the power is off, the spring pulls the plate back away from the magnet bar. The "click" is the plate slamming into the magnet as it turns on. It doesn't "click" when turned off because the plate swings away from the magnet without hitting anything.
Note: When the power is applied, the coil sets up a magnetic field in its windings. When the power is removed, the field collapses and a reverse current of high voltage will "kick back." This is called a counter electro-magnetic field (EMF) and is how your ignition coil works. If your fingers are across the coil terminals when the power is removed, you will get shocked.
Below are a couple of circuits to help understand how the relay works in real life. Relays are used to transfer high current. Many vehicles make use of the ground-to-turn-on circuit. If the relay coil terminals have battery power all the time, the ground-on circuit is how it's wired. Most horn relays are wired in the ground-on method. The steering wheel contact touches ground and turns on the horn. The horn relay is used because the 15-20 amps from the horn would arc and quickly destroy the contacts in the steering wheel.