This guy's thesis has an approachable explanation of the conjecture, as well as some of the interesting theorems that it implies (including an asymptotic version of Fermat's Last Theorem):
there's an apostrophe before the "s" in "Masters" that is stripped out by the HN code for some reason. either add that or go to http://jeffreypaulwheeler.com/ and select the link "The abc Conjecture" near the start of "Papers Section".