Question: Sorry if this is a very basic question, but I am not a physicist, but this is something that has always intrigued me. 

I have read that the properties of a black hole are its mass, spin and angular momentum. And that lower/higher the mass, higher/lower its density and so on. Although singularities are described by mathematical models, it's hard (at least for me) to believe that a singularity actually exists in the middle of space in every black hole. So why is it that singularities are actually hypothesized? Can the extremely dense nature etc. of black holes not explain its properties with a finite volume?

Also, whenever black holes are visualized in animations etc., it's always an event horizon and the observer still keeps falling through what looks like empty space. What is the significance (and the location) of the mass of the black hole itself, then, ignoring the mass it accumulates/radiates throughout its lifetime?

Answer: 
It might be worth looking at this in the context of a simpler theory of gravity - Newtons; there it is said that the gravitational attraction between two particles is inversely proportional to the square of their radial separation.

Thus the closer they are, the stronger the attraction; so what happens when they get infinitely close? We have a singularity ...

Singularities tend to signify that there is a breakdown of some kind in our theories of reality. 

Here it is easily fixed: particles cannot get infinitely close to each other - they are charged so at as certain point repulsion kicks in; another possibility is that extensionless particles are not possible - so strings; which resolves some of the infinities in a perturbation expansion ie Feynman diagrams.

The point then, is that somehow one has to resolve the singularity in a black hole; this will most probably be part of a full theory of QG - an ongoing project.

Both String theory and Quantum Loop Gravity, apparently have a 'microscopic' picture of black holes; this conjecturally explains black hole entropy - macroscopic datum - found by semi-classical means, by a detailed picture in the small; this then, suggestively resolves the singularity. 

But given the traditional conception of physics as an empirical science; the conjectural status of both these theories of QG - it can hardly be said to be entirely conclusive; but it is certainly suggestive.
The other answers have done a good job of covering the fact that we don't know if singularities actually physically exist, they might represent a breakdown of our theory, etc.

Here's another point which I think is worth making. Within the theoretical and mathematical framework of general relativity, singularities are *guaranteed* by the singularity theorems of Hawking and Penrose. Roughly, these theorems require the following things to hold:

-The spacetime you're working with satisfies certain energy conditions (we can use very weak ones to prove the singularity theorems, they're pretty physical and not very stringent).

-The spacetime satisfies certain global causality conditions (essentially guaranteeing no closed timelike curves allow time travel to the past).

-There exists a surface in the spacetime from which we could emit light and it could not escape to infinity (a formal way of testing that our spacetime contains a black hole).

If these things hold, the theorems guarantee that a singularity exists! So *within the context of general relativity*, singularities are inevitable. This was not obvious at first, and people worried that the high symmetry of our exact black hole models, and that maybe realistic black hole collapses might not have singularities. The singularity theorems well us that this is not an out. Very symmetric models do not explain away singularities, the conditions for the theorems to hold are very flexible.

Now, this guarantee requires that general relativity be a final theory which won't fail in the sort of regimes where singularities form. We think general relativity will probably have to be modified into a full theory of quantum gravity to make real predictions inside black holes, but general relativity taken literally requires singularities. It gives us reason to suspect that even if singularities don't survive in the full theory, they almost do in the sense that there will be extremely dense regions in the interiors of black holes.