What is a black hole? In this interview, which took place in November 2015, Pau Figueras, then Stephen Hawking Advanced Postdoctoral Research Fellow at the University of Cambridge, explained what black holes are, physically, how Einstein's theories predict their existence, how to describe them mathematically, and how physicists were hoping to observe them (a hope which, since this interview was recorded, has been fulfilled!).
Note: since the interview was conducted, hard evidence for black holes has been detected. It comes from gravitational waves. See Black holes exist! to find out more.