A grim reaper (GR) is a device that activates at a pre-set time. It checks if Fred--the victim--is alive. If he is, it kills him. If he isn't alive, it does nothing. For the Grim Reaper Paradox, we're supposed to imagine one GR set for 12:30, another for 12:15, another for 12:07.5, and so on. Before each time for which a GR is set, there is an earlier one. But Fred is alive alive at 12:00. Paradox ensues when we notice that Fred must be dead at 12:30 (else that 12:30 GR would have killed him), but no GR could have killed him, since if he were alive at its activation time, he would have been alive when the previous one activated, and hence would have been killed then at least.
John Hawthorne has claimed that Fred is not killed by any one GR, but by them altogether. More precisely, Fred is killed by their mereological sum.
Here's a gruesome way to see the problem with this solution. We can number the GRs in reverse: the 12:30 GR is number 1, the 12:15 one is number 2, and so on. Then suppose that the odd-numbered GRs kill by decapitating and the even-numbered ones kill by stabbing in the heart. Given the setup, Fred is either decapitated or stabbed in the heart but not both. But which one?! If he were decapitated, he would have been first stabbed. If he were stabbed, he would have been decapitated before that.