SHA-1 is a Secure Hash Algorithm (SHA) with a cryptographic strength of 80 bits. The algorithm was designed by the U.S. National Security Agency and is published by the National Institute of Standards and Technology (NIST) as a U.S. Federal Information Processing Standard. It has been widely adopted for use in many applications that require secure authentication.
The algorithm consists of a very complex set of functions, which results in a one-way hash. This allows the algorithm to provide authentication without secrecy, meaning an encrypted communication channel is not required.
A simplified schematic of how a SHA-1 authentication works.
The algorithm itself is complex, and a full explanation and documentation can be found here.