The Volokh Conspiracy
Mostly law professors | Sometimes contrarian | Often libertarian | Always independent
For the underlying story, see here:
"When I left the United States Senate, I became a professor at the University of Pennsylvania," Biden said. "And I've spent a lot of time—and the University of Delaware has the Biden School as well, so I've spent a lot of time on campus with college students."
Apparently Biden was given the title of Benjamin Franklin Presidential Practice Professor at Penn (in 2017, after the end of his Vice-Presidency), which involved him giving four speeches a year, for $390,000/year, according to PhillyMag (David Murrell).
For the underlying source of the quote, you young whippersnappers who don't know the great lines of my youth, see here.
UPDATE: Some commenters argue that Biden was technically correct to say he was a professor, given his title. But I think that misses the point: Quayle may have been technically correct when he had said "I have as much experience in the Congress as Jack Kennedy did when he sought the presidency"; Bentsen's response was that Quayle was still no Jack Kennedy, regardless of technical analogies of experience. Likewise, Biden had the title Professor (though I think it's rather misleading to describe yourself as a professor when your job doesn't involve substantial teaching or substantial research), but he's still no professor.