Features:
Time zero is an ideal limit just running the clock of the Λ-CDM model backward to the time of infinite density which is called the Big-Bang singularity.
In fact, probably no one believes there was a Big-Bang singularity. General relativity which is the basis of the Λ-CDM model and other standard cosmological models CANNOT apply at infinite density. We can run the clock of standard cosmological models back to ∼ 0.01 s with some confidence, but earlier is speculative. The observable universe tracks into Λ-CDM model at ∼ 0.01 s with some confidence.
A second meaning for time zero used in inflation cosmology, is for the time just before inflation began (see Wikipedia: Graphical timeline of the Big Bang).
The two time zeros differ by a tiny fraction of second and often they are conflated (out of laziness) as in the diagram.
The label of Big Bang in the diagram at the inflation time zero is somewhat conventional, but yours truly disagrees with this usage.
See below for what yours truly calls the Big Bang.
The observable universe on average just scales up with a(t).
By usual convention, a(t=present) = 1.
Bound systems like you, me, planetary systems, galaxies, maybe most galaxy groups, maybe galaxy clusters, and maybe some galaxy superclusters do NOT expand.
But the space between them does---this is literally a growth of space in general relativity.
The observable universe when described as a growing entity is called the expanding universe.
Once it was thought that this slowing down continue to the present.
However, since 1998 it has been believed that expansion of the universe is currently speeding up.
This feature of speeding up is incorporated in the Λ-CDM model.
The observable universe when described as having speeding up expansion is called the expanding universe.
The transition between slowing down and speeding up occurred at cosmic time ∼ 6 Gyr according to current estimates (see Wikipedia: Timeline for the formation of the Universe). However, this value may be revised significantly in the future.
This epoch started with the universe being a rather homogeneous hot gas consisting of protons, electrons, photons, neutrinos, and exotic dark matter particles (and we don't know what they are).
The density, temperature, and expansion conditions were such that nucleosynthesis cooked up the primordial abundances of light elements hydrogen, deuterium (heavy hydrogen), helium, lithium.
These primordial abundances (corrected for some evolution if needed) agree with modern-day observations to within some uncertainty. The agreement is one of the strongest pieces of evidence for the Big Bang.
At cosmic time ∼ 380,000 years, the thermodynamic conditions were right for protons and free electrons to bind together to form neutral hydrogen atoms.
This time is called recombination epoch---which is a bit of a misnomer since the protons and free electrons had never been combined before to our knowledge.
After recombination, the then existing photons mostly stopped interacting with matter forever.
These photons formed a blackbody radiation field that expanded and cooled while staying a blackbody radiation field.
It exists today as the cosmic microwave background (CMB) which is just a blackbody radiation field permeating the observable universe with a CMB T = 2.72548(57) K (Fixsen 2009) (see also Wikipedia: Cosmic microwave background radiation: Features).
The CMB is just electromagnetic radiation in the microwave band like the microwaves in your microwave oven.
Of course, some CMB does interact with matter such as when it runs into stars and and planets. Also, it interacts with matter via the gravitational fields of matter. But to a good 1st order approximation, it doesn't interact.
The CMB has a natural explanation in the Big Bang theory, and in no other viable theory.
There is other evidence too though less awesome.
At present, there is NO viable theory other than the Big Bang theory to explain the key evidence.
This is why we are fairly confident the Big Bang actually happened.
It would be astonishing if the Big Bang theory turned out to be just plain wrong. A lot of evidence would have to be explained in some totally different way.
How did this happen?
Currently, we believe about 84.5 % of matter is dark matter (see Wikipedia: Dark matter: Overview).
The dark matter is believed to be a gas of exotic particles.
The dark matter itself remained just stayed a gas in the clumps.
Dark matter does NOT interact strongly with ordinary matter, except through gravity. It may NOT interact with itself much either although there is some evidence that it repels itself at close range (see Massey et al. 2015).
There have been many generations of stars although many very old long-lived stars continue to exist.
The evolution of galaxies is immensely complex. But one key process has been growth of galaxies through galaxy mergers.
The current simulations based on the Λ-CDM model show some agreement with the observed evolution of the large-scale structure of the universe.
There are some important discrepancies too.
In future work, the discrepancies may disappear with improvements in the simulations. If they don't, the Λ-CDM model may be ruled out.
This is the realm of quantum cosmology.
We can push the clock of the Λ-CDM model back to before protons formed from quarks.
But before that, we do NOT have a consensus theory of necessary physics which is some kind of quantum field theory.
There are many theories, but which of any of them is right or at least partway right?
The inflation paradigm mentioned above gives some guidance about the necessary physics, but the guidance may no good if the inflation paradigm itself is wrong.
Inflation posits an early phase of super rapid expansion (which is the inflation itself). The super rapid expansion is illustrated in the figure.
The inflation is driven by a quantum field called the inflaton.
But there is no consensus theory for the inflaton.
There are many possibilities.
Inflation allows the universe to begin without a very special set of initial conditions.
But maybe there is some other way to dispense with very special initial conditions or maybe the initial conditions were just very special for some reason we know NOT of.
Inflation also generically predicts initial density fluctuations of the universe that are needed to develop the large-scale structure. So far those predictions are verified (see Wikipedia: Inflation: Observational Status). But there may be some other way of predicting them or a discrepancy may appear between the inflation and observations when the observations improve.
In fact, there are some people who believe the inflation paradigm is wrong or, even worse, there may be no way to prove it right or wrong---our view of the universe may be too superficial.
But cosmologists havn't given up yet.