You need to know: Prime numbers, polynomial-time algorithms, deterministic and randomised algorithms, class P.
Background: The primality testing problem is, given integer n as an input, determine whether n is prime or not.
The Theorem: On 24th January 2003, Manindra Agrawal, Neeraj Kayal, and Nitin Saxena submitted to the Annals of Mathematics a paper in which they proved the existence of deterministic polynomial-time algorithm for the primality testing problem. In other words, they proved that this problem belongs to the class P.
Short context: The primality testing problem is interesting in its own right, but also has practical applications in, for example, cryptography. In 1980, Rabin, based on earlier paper of Miller, developed an efficient randomised algorithm for this problem. However, the question whether primality testing can be done in deterministic polynomial time remained a major open problem is the field. The Theorem resolves it affirmatively.
Links: The original paper is available here. See also Section 4.10 of this book for an accessible description of the Theorem.
Theorems of the 21st century 