When reading about the "Big Bang" theories, you often here that in the beginning there was a singularity. I have read that it was a singularity at one small point in "space" or that it was everywhere at the same time.

When a singularity in this instance is talked about; a singularity of what? Energy?

The Singularity in the Big Bang theory indicates that this is some place that the mathematics can't model. It's a place (can you even use that word?) where the General Theory of Relativity breaks down. The Singularity is probably energy, but... who knows? It's something that we know absolutely nothing about. Even the best theorists don't have a clue as to what it "was".

Big Bang - Wikipedia, the free encyclopedia

There are different singularities or types of singularities. Things become infinite (mathematically) at singularities, and that's not something that's understood.

From the point of view of cosmologists infinities serve to show our ignorance. String Theory attempts to do away with infinities.

Singularity - AKA - Black hole.

The existence of which is predicted by Einsteins General Relativity.

If I understand what you are asking, then yes you are correct. The singularity is a point of energy.

Concerning your question about one small point in space, vs. everywhere at once:

Yes a singularity is an infinitely small point in space-time, however at the beginning of time there was no space so it essentially was "everywhere".