A hash algorithm turns an arbitrarily-large amount of data into a fixed-length hash. The same hash will always result from the same data, but modifying the data by even one bit will completely change the hash. Like all computer data, hashes are large numbers, and are usually written as Hexadecimal.

BitCoin uses the SHA-256 hash algorithm to generate verifiably "random" numbers in a way that requires a predictable amount of CPU effort. Generating a SHA-256 hash with a value less than the current target solves a block and wins you some coins.

Also see http://en.wikipedia.org/wiki/Cryptographic_hash