You need to know: Basic cryptography: circuits, encryption, decryption, etc.
Background: A fully homomorphic encryption scheme is a procedure that allows one to evaluate circuits over encrypted data without being able to decrypt.
The Theorem: On 31th May 2009, Craig Gentry submitted to the Proceedings of the forty-first annual ACM symposium a paper in which he showed that fully homomorphic encryption scheme exists and can be actually constructed.
Short context: The notion of a fully homomorphic encryption (FHE) scheme was introduced by Rivest, Adleman, and Dertouzos in 1978. FHE would allow to perform arbitrary computations on encrypted data without decripting them, and would therefore lead to the possibility of more secure cloud computing. For example, a program could help with preparation of tax return forms using encrypted financial information. However, the existence of FHE was a long standing open problem, and many experts believed that it does not exist. The Theorem proves that FTE exists and can be actually constructed. This makes a revolution in cryptography.
Links: The original paper is available here.
Theorems of the 21st century 