An email server that will happily send email for anyone who connects to it.

In the early days of the internet, this was useful, but these days it means that a spammer can connect to it and get it to send Spam emails to people.

No reputable company runs open relays these days as there are black lists of mail servers that are open, and many people will refuse to accept mail from listed machines. Unfortunately there are still many machines out there (especially residential users) that either have old or badly set up software (that defaulted to open relay) or have been hacked specifically to allow spammers to send mail though them.