We all know what "going online" is, so it's not too tough to figure out what someone means when they say they're "going offline." Basically, when the multiple emails back and forth between people get to be too confusing or inefficient, it's time to pick up the phone, call the person, and go offline.
Gina Trapani of BusinessWeek.com wrote a great article about the benefits of taking it offline. Sure, you don't get the dense and hassle-free transfer of information like email or instant messaging, but a phone call can clear up a misunderstanding, convey urgency and tone, and in some cases can get something done faster than through internet mediums.
One point in the article that really jumped out at me was Trapani's example of an email sitting in ten different inboxes, waiting on different responses—with some responses even waiting on other responses—and with a message chain the size of a novella.
An email like this has gone far beyond what a simple phone call could clear up. No, in this situation, and as soon as the topic goes beyond two people, it's time to start a conference call. In about the same time as it would take to type out another reply, we can start a conference with the three or ten people involved.
A ten-person conference call may last longer than the five-minute phone call needed to clear up a well-traveled email or IM miscommunication between two people… but not that much longer. And it can clear up a day's worth of emailing in relatively no time at all.
Try this experiment: next time you've got a complex situation and need to call more than one other person to straighten things out, start a conference call. Then leave a comment and tell us how it helped the situation.