For those not familiar with algebraic topology, πm(Sn) is the set of equivalence classes of continuous functions from the m-dimensional sphere to the n-dimensional sphere where two functions are considered equivalent if they are homotopic. An easy way to visualise this is that two functions are homotopic if you can interpolate a continuous animation between them. (Can you guess what industry I work in?) This set also has a group structure which is straightforward to define but which I won't go into here (unless someone requests it). That's all there is to it. How can such simplicity generate such complexity?
Monstrous Moonshine is pretty mysterious too - but it takes a lot of work to state it for a non-expert. So this wins by default.
Like John Baez I also wonder about the curious appearance of 24 on row 3.