196,884 and the Monster
The number 196,884 is the dimension of the space in which Robert Griess constructed the Monster. This representation in 196,884 dimensions splits into two: a trivial representation in 1 dimension, and an irreducible representation in 196,883 dimensions. The reason Griess chose 196,884 dimensions is that the Monster preserves a pleasing algebra structure in a space of this dimension, and the main work in his construction was to give the multiplication for this algebra, and show that it was preserved by the Monster. This algebra is now called the Griess algebra.
The number 196,884 is also the first non-trivial coefficient in the j-function in number theory, and this strange coincidence led to a connection between the Monster and the j-function, known as Moonshine. The story of these connections, and the history leading up to them, is described in my book Symmetry and the Monster.