A relay is simply an electrically controlled switch. You can look-up how an SPDT switch works, and you should be able to understand what the relay is doing.
The common and NO contacts are normally disconnected, or "off". When you apply power to the relay coil, these two contacts are connected together ("closed" or "shorted").
The common and NC work the other way. These two contacts are connected together ("on", "closed", "shorted") when the coil is not powered. When the coil is powered, these contacts open-up.
The NO & NC contacts are never connected together.
The terminology & logic is reversed from a water faucet... For a switch or relay, closed means the contacts are connected an current flows. Open means the contacts are not connected (the circuit is broken) and current doesn't flow.
Depending on your application, you may not need to use all of the contacts.
Many relays have a clear plastic cover that you can see through. I was just wiring-up this relay
about an hour ago. You can see the internal wiring and you can see the contacts opening & closing as the relay operates. (It's not for an Arduino project. This particular relay has a 12V coil, and you can't operate it directly from an Arduino.)
In my application, I'm using the common & NO contacts. The NC contacts are not connected to anything.