A cryptographic hash function is a kind of algorithm that can be run on a piece of data, often an individual file, producing a value called a checksum
Two files can be assured to be identical only if the checksums generated from each file, using the same cryptographic hash function, are identical.
Some commonly used cryptographic hash functions include MD5 and SHA-1, though many others also exist.
Note: Cryptographic hash functions are often just referred to as hash functions for short, but that's not technically correct. A hash function is a more generic term that's usually used to encompass cryptographic hash function along with other sorts of algorithms like cyclic redundancy checks.
Examples: "To verify that the latest version of Firefox I was downloading is correct, I ran a cryptographic hash function, SHA-1 to be exact, on the download and compared that checksum with the one published on Mozilla's site."