A re-engagement campaign is where you beef up your subscriber engagement (the open/read/click percentages in a sender's stats) by asking existing subscribers to click on a link to show that they're still alive, often followed by retiring/suppressing addresses who don't respond. (That suppression step boosts your engagement rates which makes you a better sender in the eyes of many ISPs, and it also helps you leave spamtrap addresses behind, so if you're having spamtrap or blacklist issues, it's very important.)
How do you do it? What should you consider? MailChimp has a solid high-level overview. Some bits are MailChimp specific, but it's still a good place to start, regardless of which ESP or email platform you use.
Need examples of compelling re-engagement content? HubSpot's got you covered, with ten great examples of effective re-engagement emails.
A site called "Essence of Email" has another dozen examples you should check out.
MailChimp also links to this Really Good Emails site which has a ton of re-engagement campaign example emails.
Note: What marketers call a re-engagement campaign, anti-spam folks would call a permission pass or reconfirmation email.