The simple thing to do is swap with another relay of the same type
(same part number). It's unlikely that you'd have two of them bad,
because some of them get harder use than the others, and most have two
sets of contacts but only use one.

For example, the A/C relay only uses the normally closed contacts,
while the fuel pump relay only uses the normally open contacts. So when
you swap them, you're using contacts that never drew any current.

The usual fail mode is that the contacts get burned from repeatedly
switching on and off, and no longer conduct well.

You can also have the solenoid go bad, or the mechanical part stick or
break.

If you have a pinout of the relay and an ohmmeter, you can measure the
resistance of the normally closed contacts, should be zero ohms. Then
connect the relay coil (solenoid) across your battery to energize it
(you should hear it click), and measure the normally open contacts,
should also be zero ohms.  

-Paul