The term moonshine is not well-defined. A moonshine theory involves finite simple groups in some way and expresses a coincidence or connection with some relatively distant area of mathematics or physics. Such a theory when first proposed may have been surprising if it involved a leap across a great intellectual distance.

Below are the theories I recalled (2 February, 2010). Please inform me if I have left off a good moonshine type theory or basic reference. There is a lot of material on the internet, too.

Monster and modular forms on the complex upper half plane (late 1970s) by McKay, Thompson, Conway, Norton

MR0554399 (81j:20028) Conway, J. H.; Norton, S. P. Monstrous moonshine. Bull. London Math. Soc. 11 (1979), no. 3, 308--339. (Reviewer: Stephen D. Smith) 20D08 (10D12)

The extended $E_8$-diagram and $2A$-dihedral subgroups of the monster (there are versions of this for subdiagrams of the extended $E_8$-diagram and other finite groups) McKay; Glauberman and Norton.

MR0604577 (82e:20014) McKay, John Graphs, singularities, and finite groups. The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), pp. 183--186, Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, R.I., 1980. 20C99 (05C25 14B05 17B10)

MR1877755 (2002k:20024)
Glauberman, George(1-CHI); Norton, Simon P.(4-CAMB)
On McKay's connection between the affine $E_8$ diagram and the Monster. Proceedings on Moonshine and related topics (Montréal, QC, 1999), 37--42,
CRM Proc. Lecture Notes, 30, Amer. Math. Soc., Providence, RI, 2001.
20D08 (20F55)

The extended Dynkin-diagrams and finite subgroups of $SU(2,C)$ via Cartan matrices

MR0604577 (82e:20014) McKay, John Graphs, singularities, and finite groups. The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), pp. 183--186, Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, R.I., 1980. 20C99 (05C25 14B05 17B10)

MR0647198 (83j:20046)
Drucker, Daniel; Frohardt, Daniel
Irreducible root systems and finite linear groups of degree two.
Bull. London Math. Soc. 14 (1982), no. 2, 142--148.
20G05 (14J17 20C30 22E60)

appendix to the book of P. Slodowy [Simple singularities and simple algebraic groups, Lecture Notes in Math., 815, Springer, Berlin, 1980; MR0584445 (82g:14037)]

The extended Dynkin-diagrams and finite subgroups of $SU(2,C)$ via tensoring

MR0604577 (82e:20014) McKay, John Graphs, singularities, and finite groups. The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), pp. 183--186, Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, R.I., 1980. 20C99 (05C25 14B05 17B10)

The Norton conjectures about commuting pairs of Monster elements by Simon Norton

Proc. Sympos. Pure Math. 47 Part 1,
The Arcata Conference on Representations of Finite Groups (Arcata, Calif.,
1986), 209–210, Amer. Math. Soc., Providence, RI (1987)

The Moonshine Vertex Operator Algebra, constructed by Frenkel, Lepowsky, Meurman

MR0996026 (90h:17026) Frenkel, Igor; Lepowsky, James; Meurman, Arne Vertex operator algebras and the Monster. Pure and Applied Mathematics, 134. Academic Press, Inc., Boston, MA, 1988. liv+508 pp. ISBN: 0-12-267065-5 (Reviewer: Kailash C. Misra) 17B65 (17B67 20D08 81D15 81E40)

Modular Moonshine (positive characteristic moonshine), by Borcherds and Ryba

MR1390654 (98b:17030) Borcherds, Richard E.; Ryba, Alex J. E. Modular Moonshine. II. Duke Math. J. 83 (1996), no. 2, 435--459. (Reviewer: Koichiro Harada) 17B69 (11F22 20D08)

MR1372729 (97c:20022) Ryba, A. J. E. Modular Moonshine? Moonshine, the Monster, and related topics (South Hadley, MA, 1994), 307--336, Contemp. Math., 193, Amer. Math. Soc., Providence, RI, 1996. (Reviewer: A. S. Kondrat?ev) 20C34 (11F22 17B67 20D08)