In order to proper understanding of Space-Time curvature, we should read about General theory of relativity.
General theory of relativity
According to General theory of relativity, massive object wrap the space time around them. This happens due to having of gravity of massive objects. The objects these have mass wrap space-time around them.
Geometry of universe is related with mass of the objects. It is a common question is that what is the structure of universe. Is it open or closed? or Is it flat in structure. These are determined by the mass density of universe.
When the density of universe is equal to the critical density, then the universe is zero curvature.