A Relay is essentially any system that accepts and forwards mail to SMTP domains for which it does not act authoritatively. That's a mouthful - what does that mean exactly? Let's say you address a message to email@example.com and send it to my corporate server. If my server accepts and forwards the message, my server is a relay. If my server accepts and forwards all messages from everyone, it's an open relay. Open relays are considered bad because Spammers find and utilize them to help disguise the source of the spam. Being an Open Relay is bad because recipient systems will see the spam coming from your system and will quickly degrade your reputation. You can end up on blacklists then have your legitimate messages rejected.
A system can be a Relay without being an Open Relay. A Relay is not a bad thing in itself.