SHA-256 is covered in full under SHA256, which explains how the algorithm works and where it is used. This page covers the hyphenated spelling, which is the form used in the official US standard and in most security documentation.
SHA-256 is a cryptographic hash function in the SHA-2 family, designed by the US National Security Agency and published by the National Institute of Standards and Technology in 2001. It takes data of any length and produces a fixed 256-bit output, usually shown as 64 hexadecimal characters. Changing even one character of the input produces a completely different output, and it is considered infeasible to find two inputs with the same hash or to work backwards from a hash to its input.
Bitcoin relies on SHA-256 in several places. Miners repeatedly hash block headers, applying SHA-256 twice, in search of a value below the network's target, the core of proof of work. Hashes also link each block to the previous one and build the Merkle tree of transactions. Specialised ASIC machines are built to compute SHA-256 at enormous speed.
Outside crypto, SHA-256 is used in website certificates, software signing and password systems. The spellings SHA-256, SHA256 and "SHA 256" all refer to the same function.