Homomorphic encryption is a type of encryption which allows one to manipulate data without revealing it what the data truly is. The resulting computations, if decrypted, would be the same had the computations been performed on unencrypted data.
Homomorphic encryption is useful for privacy preservation for when you outsource data storage, especially when storing sensitive data. It has the potential to make cloud computing more secure, since we no longer have to expose sensitive data to perform operations on it.
For example: a homomorphically encrypted financial database could query how much an employee made in the second quarter. However, none of the information about the employee would be exposed.
Currently, fully homomorphic encryption requires immense computational power to perform even simple tasks — rendering it effectively theoretical (for now). It might take a decade for the scheme to become practical.