How did this 'black fabric' that we call space start? How was it made? How was the initial singularity formed? How did the first atom come into existence?
If it all started from a tiny particle that expanded and is now still expanding, how did that tiny particle come into existence ?
I'd like to add onto what James K said, but I'll try and simplify things too.
A singularity may sound like everything was at a "single point", but that's not what the word means in physics/math. A singularity, loosely speaking, is just a point where a function reaches infinity. It's not a physical object, but an input to a function. As James mentioned, 1/x has a singularity at x=0. At the moment of the Big Bang, several functions have singularities, as I discuss below.
So with that set aside, let's explain the Big Bang. Our modern theory of gravity, general relativity, indicates that the universe is currently expanding (i.e. space is being created between all matter). By that logic, we expect the matter in the universe to be closer together as we look further back in time. Our physical theories indicate that at earlier periods, temperatures were so high that even atoms couldn't form.
Our theories can handle these conditions pretty well, but eventually they start to fall apart. At a point around 13.8 billion years in the past, our calculations predict that the amount of space in the Universe should be zero. As a result, density, gravity and temperature would skyrocket to infinity — in other words, they have singularities at that moment. This moment is known as the Big Bang.
These infinities are a problem, since our theories break down at such extreme conditions. The Big Bang is an unsolved problem that modern physics has yet to explain. It's not the beginning of the universe per se, but rather we do not know what happened around, or before, that moment.
Your confusion comes from the words "singularity" and perhaps "atom" and "particle"
An equation like y=1/x is singular when x=0. For a physical object modelled by the equation y=1/x the model can't be applied to understand the object at x=0, and it may or may not make sense to use the model for negative values of x.
The singularity is not an object. It would be better if people said "the equations that model the universe become singular at a point in time. The models that we have for the universe don't work at time t = 0, and it is not clear if negative values of "t" are physically meaningful.
The singularity was not a tiny particle that expanded. Since the singularity is not an object, it doesn't make much sense to ask "how did it form"
Atoms are another thing altogether. We know with great detail how protons and electrons formed. How most of the protons that formed were annihilated by antiprotons, but a few remained, and as the universe cooled, electrons bound with the protons to form atoms.
I guess here there is an assumption that there was a singularity, where as Sir Circumference points out, both relativity and quantum mechanics break down.
Einstein even considered that the universe goes through eternal oscillations of expansion, crunch and bounce before Tolman showed that entropy can only increase, thereby resulting in progressively slower expansion. Which means through extrapolation there must have been some sort of beginning. Entropy is the problem in any cyclical model of the universe that involves matter.
Recent theorists such as Paul Steinhardt (Princeton) and Neil Turok (Cambridge) have proposed another theory of cyclical epochs where there is a limit on the net expansion phase brought about by colliding branes (membranes) and dark energy limiting entropy.
An alternative theory is the Baum-Frampton Cyclic Model which also proposes cyclical phases in regard to expansion, turnaround, contraction and bounce. Expansion results in the fragmentation of all matter into patches, or separate universes. Just before the end of time, at the turnaround, the patches individually contract and then bounce to infinitely repeat the cycle. Entropy becomes a non-player as the theory contends that contraction of each universe results in an empty state, that is, only dark energy and no matter.
Steinhardt–Turok model https://arxiv.org/abs/hep-th/0111030
Baum-Frampton model https://arxiv.org/abs/hep-th/0703162
