The Monster group $M$ is a finite group that is the largest of the sporadic finite simple groups. It has order
and contains all but six, the ‘pariahs’, of the other 25 sporadic finite simple groups as subquotients.
See also Moonshine.
The Monster group was predicted to exist by Bernd Fischer and Robert Griess in 1973, as a simple group containing the Fischer groups? and some other sporadic simple groups as subquotients. Subsequent work by Fischer, Conway, Norton and Thompson estimated the order of $M$ and discovered other properties and subgroups, assuming that it existed. In a famous paper
Griess proved the existence of the largest simple sporadic group. The author constructs “by hand” a non-associative but commutative algebra of dimension 196883, and showed that the automorphism group of this algebra is the conjectured friendly giant/monster simple group. The name “Friendly Giant” for the Monster did not take on.
After Griess found this algebra Igor Frenkel, James Lepowsky and Meurman and/or Borcherds showed that the Griess algebra is just the degree 2 part of the infinite dimensional Moonshine vertex algebra.
There is a school of thought, going back to at least Israel Gelfand, that sporadic groups are really members of some other infinite families of algebraic objects, but due to numerical coincidences or the like, just happen to be groups (see this nCafe post). One version of this, in the case of the Monster (and perhaps for other sporadic groups via Moonshine phenomena) is that what we know as the Monster is just a shadow of a 2-group, as the Monster can be constructed as an automorphism group of a conformal field theory, a structure rich enough to have a automorphism 2-group(oid) (see this nCafe discussion).
The Monster admits a reasonably succinct description in terms of Coxeter groups. Let $[n]$ denote the linear graph with vertices $0, 1, \ldots, n$ with an edge between adjacent numbers $i, i+1$ and no others. If $1$ is the terminal (1-element) graph, there is a map $0: 1 \to [n]$, mapping the vertex of $1$ to the vertex $0$. Regarding this as an object in the undercategory $1 \downarrow Graph$, let $Y_{443}$ be the coproduct of the three objects $0: 1 \to [4]$, $0: 1 \to [4]$, $0: 1 \to [3]$ in $1 \downarrow Graph$. This (pointed) graph has 12 elements and is shaped like a $Y$, with arms of length 4, 4, 3 emanating from a central vertex of valence $3$.
Regard $Y_{443}$ as a Coxeter diagram. The associated Coxeter group $C_{443}$ is given by a group presentation with 12 generators (represented by the vertices) of order $2$ (so 12 relators of the form $x^2 = 1$), with a relation $(x y)^3 = 1$ if $x, y$ are adjacent vertices (so 11 relators, one for each edge), and $x y = y x$ if $x, y$ are non-adjacent (55 more relators). This Coxeter group (12 generators, 78 relators) is infinite, but by modding out by another strange ‘spider’ relator
the resulting quotient $N$ turns out to be a finite group. Here $a$ is the central vertex of valence $3$, $b_1, c_1$ are on an arm of length $4$ with $b_1$ adjacent to $a$ and $c_1 \neq a$ adjacent to $b_1$; similarly for $b_2, c_2$ on the other arm of length $4$, and for $b_3, c_3$ on the arm of length $3$. See here if this is not clear.
It turns out that $N$ has a center $C$ of order $2$, and the Monster $M$ is the quotient, i.e. the indicated term in the exact sequence
This implicitly describes the Monster in terms of 12 generators and 80 relators.
Such “$Y$-group” presentations (Coxeter group based on a similar $Y$-diagram, modulo a spider relation) are linked to a number of finite simple group constructions, the most famous of which is perhaps $Y_{555}$ which is a presentation of the “Bimonster” (the wreath product of the Monster with $\mathbb{Z}/2$). See Ivanov for a general description of these. The presentation of the Monster given above was established in Ivanov2.
Adam P. Goucher (http://mathoverflow.net/users/39521/adam-p-goucher), Presentation of the Monster Group, URL (version: 2013-09-15): http://mathoverflow.net/q/142216
Alexander Ivanov, Y-groups via transitive extension, Journal of Algebra, Volume 218, Issue 2 (August 15, 1999), 412–435. (web)