What is asymmetric cryptography?

Whitfield Diffie and Martin Hellman were jointly awarded the 2015 ACM A.M. Turing Award today.  Their 1976 paper, New Directions in Cryptography, essentially created asymmetric cryptography.  Today, asymmetric cryptography secures our online communications—from PGP-secured texts, emails, and files, to TLS and SSL-secured websites (including this one).  So how does asymmetric cryptography work, and how is the Diffie-Hellman key