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What is spacetime geometry?

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