It is (to date) the central object in monstrous moonshine, since its character is the $q$-expansion of the modular $J$-function, its automorphism group is the monster simple group, and the graded trace of any element of the monster is the $q$-expansion of a genus zero modular function. The construction of this structure (by Frenkel, Lepowsky, and Meurman) involves ascending a hierarchy of objects that are by themselves quite intricate and beautiful.
One begins with the extended binary Golay code of length 24. Up to symmetries, it is the unique copy of $\mathbb{F}_2^{12}$ in $\mathbb{F}_2^{24}$, for which any five basis vectors are contained in a unique codeword (i.e., it forms a Steiner $(5,8,24)$ system). The codewords are separated by Hamming distance at least 8, so even if 3 bits in a code word are changed the error can be corrected. The automorphism group of the Golay code is the sporadic simple group $M_{24}$ of order 244823040.
Using the Golay code to produce coordinates of generators, one constructs the Leech lattice $\Lambda$, which is a rather densely packed copy of $\mathbb{Z}^{24}$ in $\mathbb{R}^{24}$. One can also make the Leech lattice as a subquotient of the even unimodular lattice $I\!I_{25,1}$, which has its own exceptional properties. Peter Shor mentioned the Leech lattice in another answer, so I'll just note that its automorphism group is a double cover of Conway's sporadic simple group $Co_1$, which has order 4157776806543360000.
For any positive definite even lattice $L$, there is a canonical construction of a vertex operator algebra graded by that lattice, called the lattice vertex algebra $V_L$. I think physicists say that it is the algebra of chiral symmetries of a conformal field theory describing a bosonic string propagating in the torus $L \otimes \mathbb{R}/L$ (but I may have mixed up the words). It has an action of the holomorph of the algebraic torus $L \otimes \mathbb{C}^\times$.
The "-1" automorphism of the Leech lattice induces an automorphism $\theta$ of $V_\Lambda$, and there is a unique irreducible $\theta$-twisted module $V_\Lambda(\theta)$ that inherits an action of the centralizer $2^{1+24}.Co_1$ of $\theta$ in the automorphism group of $V_\Lambda$. The monster vertex algebra is formed by taking the direct sum of fixed points: $V^\natural = (V_\Lambda)^\theta \oplus (V_\Lambda(\theta))^\theta$.
Apparently, the hard part was proving that the monster acts on $V^\natural$ by automorphisms.
There are some additional conjectural reasons for considering it beautiful:
In the same paper where it was constructed, it was conjectured to be the unique vertex operator algebra with central charge 24, character equal to the modular $J$ function, and representation category equivalent to $Vect$. (Naturally, this does not account for higher structure like twisted modules.)
Witten suggested that it is dual to pure 3-dimensional quantum gravity with minimal cosmological constant by AdS/CFT correspondence.
