Moonshine usually refers to the mysterious connections between the Monster simple group? and the modular function $j$, the j-invariant. There were a bunch of conjectures about this connection that were proved by Richard Borcherds, en passant mentioning the existence of the Moonshine vertex algebra (constructed then later by FLM). Nowadays there is also Moonshine for other simple groups, by the work of J. Duncan. Everntually there shoould be an entry for the general moonshine phenomenon.
The whole idea of moonshine began with John McKay’s observation that the Monster group’s first nontrivial irreducible representation has dimension 196883, and the j-invariant $j(\tau)$ has the Fourier series expansion
where $q=\exp(i2\pi\tau)$, and famously 196883+1=196884. Thompson observed in (1979) that the other coefficients are obtained from the dimensions of Monster’s irreducible representations.
But the monster was merely conjectured to exist until Griess (1982) explicitly constructed it. The construction is horribly complicated (take the sum of three irreducible representations for the centralizer of an involution of…).
Frenkel, Lepowsky, Meurman (1984) construct an infinite-dimensional module for the Monster vertex algebra. This is by a generalized Kac-Moody algebra via bosonic string theory and the Goddard-Thorn "No Ghost" theorem. The Monster group acts naturally on this “Moonshine Module” (denoted by $V\natural$).
To cut the story short, we end up getting from the Monster group to a module it acts on which is related to “modular stuff” (namely, the modular $j$ function). The idea Terry Gannon pitches is that Moonshine is a generalization of this association, it’s a sort of “mapping” from “Algebraic gadgets” to “Modular stuff”.
John Conway and Simon Norton, “Monstrous moonshine.” Bull. London Math. Soc. 11 (1979), no. 3, 308–339; MR0554399 (81j:20028)
Igor Frenkel, James Lepowsky, Arne Meurman, “A natural representation of the Fischer-Griess Monster with the modular function $J$ as character.” Proc. Nat. Acad. Sci. U.S.A. 81 (1984), no. 10, Phys. Sci., 3256–3260. MR0747596 (85e:20018)
Robert Griess, “The friendly giant.” Invent. Math. 69 (1982), no. 1, 1–102. MR671653 (84m:20024)
John G. Thompson, “Some numerology between the Fischer-Griess Monster and the elliptic modular function.” Bull. London Math. Soc. 11 (1979), no. 3, 352–353. MR0554402 (81j:20030)