Singularity in math typically refers to a point at which a function becomes infinite. This often occurs due to a "division by zero".
Consider the gravitational attraction of two point masses:
F = G*m1*m2/r^2
where G is the gravitational constant, m1 and m2 are the masses, and r is the radius between the masses.
As the two point masses become closer and closer together (as in a black hole), the radius goes to zero, and the force becomes infinite. This represents a singularity. In fact, the center of a black hole is often called a singularity for this reason