We can think of a relay as an electromagnetic switch. Apply a voltage to the coil and a magnetic field is generated. This magnetic field sucks the contacts of the relay in, causing them to make a connection. These contacts can be considered to be a switch. They allow current to flow between 2 points thereby closing the circuit.

Let's consider the following example. Here we simply turn on a bell (Lunch time!) whenever a switch is closed. We have 3 real-world parts. A switch, a relay and a bell. Whenever the switch closes we apply a current to a bell causing it to sound.

Notice in the picture that we have 2 separate circuits. The bottom(blue) indicates the DC part. The top(red) indicates the AC part.

Here we are using a dc relay to control an AC circuit. That's the fun of relays! When the switch is open no current can flow through the coil of the relay. As soon as the switch is closed, however, current runs through the coil causing a magnetic field to build up. This magnetic field causes the contacts of the relay to close. Now AC current flows through the bell and we hear it. Lunch time!

A typical industrial relay