From the ABC@Home Website:

What is ABC@home?

ABC@home is a distributed computing project enabling a great search for so called abc-triples. The project is comparable in a way to GIMPS, another mathematical project. These abc-triples are positive integers a,b,c such that a b=c, a < b < c, a,b,c have no common divisors and c > rad(abc), the so-called radical of abc. The ABC conjecture says that there are only finitely many a,b,c such that log(c)/log(rad(abc)) > h for any real h > 1. The ABC conjecture is currently one of the greatest open problems in mathematics. If it is proven to be true, a lot of other open problems can be answered directly from it. More information can be found here .

Who is involved?

The Mathematical Institute of Leiden University together with Kennislink, a dutch institute to promote science. The science group that constructed a new algorithm to find these abc-triples consists of prof. HW. Lenstra jr, dr. B. de Smit, drs. WJ. Palenstijn. Lenstra and de Smit previously worked on completing a lithograph of Escher.

Why?

It is one of the greatest open mathematical questions, one of the holy grails of mathematics. It will teach us something about our very own numbers.