The Clappers are a band of suicide bombers who don't live for the applause, they die for it. We know very little about them. They're the Occupy Main Street of terrorist movements, and have no clear message: "The threat of clappers is so effective because no one knows what they really stand for" (2.19.7). See what we mean?
They're a vague threat early on, but the threat becomes real when Lev joins a group of them—Mai, Blaine, and Cleaver. Cleaver is a pilot who plans on betraying the Admiral. Blaine has tattoos (and in this world, tattoos = bad news) and Mai, like Lev, has expected to die for a long time because "Back in China […] people were killing off their baby girls left and right" (2.19.92). Death is kind of these teens' thing.
These people live for "chaos" and they "mess with the world" (5.35.42), which is exactly what happens in the book's conclusion. Their explosions change things in ways Lev never anticipated. Check out our "What's Up With the Ending?" section for more.