Think of a very large ball. Even though you
look at the ball in three space dimensions, the outer surface of the
ball has the
geometry of a sphere in two dimensions, because there are only
two independent directions of motion along the surface. If you were
very small and lived on the surface of the ball you might think you
weren't on a ball at all, but on a big flat two-dimensional plane. But
if you were to carefully measure distances on the sphere, you would
discover that you were not living on a flat surface but on the curved
surface of a large sphere.
The idea of the curvature of the surface of
the ball can apply to the whole Universe at once. That was the great
breakthrough in Einstein's theory of general
relativity. Space and time are unified into a single geometric
entity called spacetime, and the spacetime
has a geometry, spacetime can be curved
just like the surface of a large ball is curved.
When you look at or feel the surface of a large
ball as a whole thing, you are experiencing the whole space of a sphere
at once. The way mathematicians prefer to define the surface of that
sphere is to describe the entire sphere, not just a part of it. One
of the tricky aspects of describing a spacetime geometry is that we
need to describe the whole of space and the whole of time. That means
everywhere and forever at once. Spacetime geometry
is the geometry of all space and all time together as one mathematical
entity.

What determines spacetime geometry?

Physicists generally work by looking for the
equations of motion whose solutions best describe the system they want
to describe. The Einstein equation is
the classical equation of motion for spacetime.
It's a classical equation of motion because quantum behavior is never
considered. The geometry of spacetime is treated as being classically
certain, without any fuzzy quantum probabilities. For this reason, it
is at best an approximation to the exact theory.
The Einstein equation says that the curvature
in spacetime in a given direction is directly related to the energy
and momentum of everything in the spacetime that isn't spacetime itself.
In other words, the Einstein equation is what ties gravity to non-gravity,
geometry to non-geometry. The curvature is the gravity, and all of the
"other stuff" -- the electrons and quarks that make up the
atoms that make up matter, the electromagnetic radiation, every particle
that mediates every force that isn't gravity -- lives in the curved
spacetime and at the same time determines its curvature through the
Einstein equation.

What is the geometry of our spacetime?

As mentioned previously, the full description
of a given spacetime includes not only all
of space but also all of time.
In other words, everything that ever happened and ever will happen in
that spacetime.
Now, of course, if we took that too literally,
we would be in trouble, because we can't keep track of every little
thing that ever happened and ever will happen to change the distribution
of energy and momentum in the Universe. Luckily, humans are gifted with
the powers of abstraction and approximation,
so we can make abstract models that approximate
the real Universe fairly well at large distances, say at the
scale of galactic clusters.
To solve the equations, simplifying assumptions
also have to be made about the spacetime curvature. The first assumption
we'll make is that spacetime can be neatly
separated into space and time. This isn't always true in curved
spacetime, in some cases such as around a spinning black hole, space
and time get twisted together and can no longer be neatly separated.
But there is no evidence that the Universe is spinning around in a way
that would cause that to happen. So the assumption that all of spacetime
can be described as space changing with time
is well-justified.
The next important assumption, the one behind
the Big Bang theory, is that at every time in the Universe, space
looks the same in every direction at every point. Looking the
same in every direction is called isotropic, and looking the same at
every point is called homogeneous. So we're assuming that space is homogenous
and isotropic. Cosmologists call this the assumption of maximal
symmetry. At the large distance scales relevant to cosmology,
it turns out that it's a reasonable approximation to make.
When cosmologists solve the Einstein equation
for the spacetime geometry of our Universe, they consider three basic
types of energy that could curve spacetime:
1. Vacuum energy
2. Radiation
3. Matter
The radiation and matter in the Universe are treated like a uniform
gases with equations of state that relate pressure to density.
Once the assumptions of uniform energy sources
and maximal symmetry of space have been made, the Einstein equation
reduces to two ordinary differential equations that are easy to solve
using basic calculus. The solutions tell us two things: the geometry
of space, and how the size of space
changes with time.