You know what a prime number is, right? It's a number divisible only by itself and the number one. The first several prime numbers are 2, 3, 5, 7, 11, and 13.
It was Euclid who showed us that there are infinitely many primes. The proof is that if you take all the prime numbers known, multiply them all together and add one, then you end up with a new prime number and one a great deal larger than the primes used to find it. The largest prime number so far discovered is 7,816,230 digits long! Even if you could write super fast, say ten digits a second, and keep up that pace for days without resting, it would still take you nine days to write out the number.
To find very large prime numbers is tricky business. To ensure that a number is indeed a prime, one has to check that none of the many numbers that come before it will divide evenly into it. Even to do this math for a number just seven digits long would take the average human quite a bit of time. Mathematicians have short cuts that speed up the process, but those short cuts go only so far when we're talking about really, really big numbers.
That's where the Great Internet Mersenne Prime Search comes in. All over the world, people are using their computers to search for larger and larger prime numbers. These computers are programmed to search for large primes whenever they're not busy with other tasks. This latest record holder was discovered by the computer of a German eye surgeon, who also happens to be passionate about math.