Wei refers to the smallest denomination of ether (ETH), the currency used on the Ethereum network. For instance, 1 ether (ETH) is equivalent to 1x1018 wei (which may also be represented in exponential notation: 1e18 wei).
In this context, wei and other denominations of ether may be useful for describing small value transactions, especially in regards to very small amounts of ETH that are typically used for transaction fees or when setting the gas limit of a transaction.
As presented on the Ethereum technical documents, there are several other denominations of ether, each representing a multiple of wei. Gwei and ether are among the most used denominations.
Similarly, Bitcoin uses the term satoshi to describe the smallest unit of its currency. 1 satoshi is equivalent to 0.00000001 BTC (or 10-8 BTC).