Hash

A hash is simply a number that is created by performing a hashing algorithm on a file or a message. No matter how many times the hashing algorithm is calculated, it will always return the same number - unless the file or message has been modified.
When used in this context, a hash provides integrity. The hash is calculated at the source, and then again at the destination. If the hashes are different, the file or message has lost integrity.
However, what if someone could modify the message enough so that the new has is the same as the original hash. It would look like it has not lost integrity because the hashes are the same, but it has lost integrity. A secure hash (one of sufficient strength) cannot be recreated. In other words, someone should not be able to modify a file or message enough to reproduce the original hash.