A singularity is any place in any kind of physics where a quantity (described by the mathematics of the physics) approaches infinity.

Now, from what I can recall: In some cases, the quantity is one that we can ignore. That is, it is a quantity that doesn't play a role in determining the dynamics of the system at that point. For example, in tracing a wave that converges on a single point (like say a wave travelling through a drum head and bouncing back to the middle), certain quantities of the wave will approach infinity as we mathematically approach the point of convergence, but none of these quantities are be used to determine the dynamics of the wave.

In the case of cosmology, which I am more sure of, if we trace the expansion of the universe backwards, we get a number of singularities. One of these is the average density of the universe. This is one among other cosmological quantities that behave similarly poorly at the same point in the past and that are important quantity to the dynamics of the universe.

This would be a big problem for cosmology, if not for the fact that the conditions that allow us to describe the universe using our best theories break down near where we expect there to be a singularity in the past. So there might not actually be a singularity there, or there might not be a singularity there that is fatal to cosmological theories.

Attempts to work out alternatives around this singularity really require significantly more information about high energy physics.