In General
Transforms a set of characters into a fixed-length value that represents the original. It involves a one-way, irreversible function that converts the input parameter into a randomly computed string. A good hashing function should never produce the same result (collision) from different inputs.
In Databases
Hashing allows faster data retrieval by assigning the original value or key to a hash table indexed by its hash value. The use of the computed hash value makes searching faster compared to a per letter lookup. Hashing performs best when data is discrete and random.
In Security
Hashing's goal in security is to create irreversible data that can only be reproduced by trying a huge set of inputs. Hashing is very useful when storing sensitive information that are one-way in nature like passwords and social security numbers. Although it has connections with encryption, it should not be confused with it.
In File Systems
Hashing can be used to verify the contents of a file. When a file has been hashed, it can be used as a point of comparison to determine if it has been changed or corrupted.