The Right to Vanish is a name given to a couple of related ideas about online identities.
The first part of it is the right to walk away from a pseudonym. For example, I could close down this site tomorrow, leaving no explanation as to my actions, and no one would be able to hold me to account for it. That's just the way it is. People can delete their LiveJournal accounts, their DeviantArt accounts, and so on. Things I posted elsewhere, of course, would mostly still exist.
The other meaning of the idea is that someone retains the ability to retract anything they've said online at anytime. If someone really wanted to vanish, the foolproof way is to delete anything they have access to that they created. This is currently mostly impossible in most social networks. Hell, back when everyone used Usenet, no one suspected that something like Google Groups would come along and archive everything they ever said. Since many people used their real names (or attached a specific pseudonym to a real-life identity), their Right to Vanish has been permanently removed. The content of these posts continues to be attached to their names, even though their opinions may have changed in the intervening years or decades.
I think I've given up my right to vanish at this point, because although it's not explicit, most of my actions on the internet can be pieced together into one quilt with some work.