Relays allow the control of a high current circuit using a low current circuit, it is essentially a fancy, remotely operated switch. Using a low current circuit as the control means that the control circuit and associated components do not need to be big and heavy duty to take all the current of the equipment being controlled.
Lets take a look at a typical relay and the circuit diagram (normally shown on the side). This relay is from the Porsche and operates the two radiator fans.
On the circuit diagram shown on the left, between terminals 86 and 85 is a representation of a coil that induces a magnetic field when it is charged. The components of the relay are designed to be light and operate easily so the coil is considered to be on the low current circuit. When pins 86 and 85 are connected to a 12V supply, the magnetic field in the coil attracts a metal piece on the arm that is attached to levers A and 30. This causes the arm to move towards the coil and close the switches completing the heavy duty circuits and powering the fans.
So there isn’t a lot that can go wrong with relays, since the switch is the only moving part, that is the first place on my list to start troubleshooting. On most relays you can hear the switch(es) click when the actuating circuit is charged, so step 1 is to connect it to a 12V supply. On our diagram above we would connect pin 86 to the positive battery terminal and touch pin 85 on a ground and listen for the click – you might also be able to feel the switch closing through the case. If you can’t hear anything, you can check the high current circuit continuity. Connect pins A and B to a multi-meter set on continuity or resistance, now when you touch pin 86 to ground, you should see a reading on the multi-meter indicating that the circuit is closed (i.e. less than infinite resistance). Repeat for the second high current circuit using pins 30 and 87. If there is still no connection on the high current circuit and you’re sure that the low current circuit is connected properly, the relay is done. You can open it up and have a look but the most common failures are likely to be mechanical issues with the tiny switch or a short in the induction coil, unfortunately, a new relay is likely to be required.
There are several other styles of relays, most have only one high current circuit, but all operate on the same principals. If you have the circuit diagram, you should be able to work out which pins go to the coil to activate the switch(es) and thus test the functionality of the relay.