'Gas fee' is the term for the inherent costs of making a transaction of cryptocurrency, originating and used primarily in making transactions in the Ethereum network (at this time, the majority of cryptocurrencies that you might encounter will be based on Ethereum).
Gas fees may be viewed simply as a transaction fee for most users, but it is a bit more complicated; the gas fee is calculated as the gas limit times the gas price. The price is exactly what it sounds like, the price paid in Ether (ETH) per operation. The limit is the maximum gas the user is willing to spend to complete the transaction. This serves two functions; this sets up a market, where those paying higher gas fees may be moved to the front of the queue, and provides a maximum past which the transaction will short-circuit, preventing infinite loops and other computation errors and abuses.
While these fees are in fact fees with real monetary value, they are commonly calculated in Gwei -- a Gwei is 10-9 Ether -- and they are generally very low.