Equihash is a Proof-of-Work (PoW) mining algorithm developed by Alex Biryukov and Dmitry Khovratovich at the University of Luxembourg. The idea was proposed in an academic paper: Equihash: Asymmetric Proof-of-Work Based on the Generalized Birthday Problem.
'Asymmetric' in this instance means that the PoW algorithm requires significantly less memory and time for proof verification than it does for proof computation. In other words, it's difficult to compute but easy to verify.
Miners of the cryptocurrencies that use Equihash need to solve an Equihash puzzle using Wagner's algorithm for the Generalized Birthday Problem (GBD) in order to propose a block to add to the end of the blockchain.
A Generalized Birthday Problem
An analysis of Zcash's use of the Equihash proof-of-work scheme
Alexander Peslyak (aka Solar Designer)
Equihash 144,5 - The New Algorithm That Requires 2.5GB of Memory
June 12th, 2018
Equihash: Asymmetric Proof-of-Work Based on the Generalized Birthday Problem (Full version)
Alex Biryukov, Dmitry Khovratovich
Why Equihash? - Zcash
Zooko Wilcox and Jack Grigg