Wei is the smallest denomination of ether, the cryptocurrency coin used on the Ethereum network.
1 Ether = 1,000,000,000,000,000,000 Wei (1018)
As the prices of various cryptocurrencies, including ether, have skyrocketed during the last year, transaction sizes have become smaller.
To correctly denote the quantity of transactions which may appear a very small fraction in terms of ether but a high value when converted to U.S. dollars or other real world currency, new units have been created to correctly identify and support the transactions.
Wei is named after Wei Dai, a cryptography activist who is known for supporting widespread use of strong cryptography and privacy-oriented technologies.