If you want to solve equations of motion to describe the time evolution of a system, either classically or quantum mechanically, you need to impose initial condition at one point in time, and then under some conditions the entire evolution of the system (forward and backwards) is determined. This is the type of things physicists do all the time.
Now, general relativity is a theory of spacetime, so it is not clear that any spacetime manifold will have well-defined evolution of the sort I described, where the conditions at a spatial slice at one point in time (called Cauchy surface) determines the system everywhere. For that to be true there has to be a way to separate what is the time direction at every point in spacetime.
If this can be done you express the spacetime as a series of spatial slices which evolves in time (called foliation of spacetime), and you have now a problem which amounts to describing how those spatial slices evolve, which is a traditional initial value problem which physicists know and love. Manifolds for which this can be done are called globally hyperbolic, and those are the ones which are easier to discuss, though there are well-known examples of spacetimes which are not globally hyperbolic.
Once you find one way to do it, one "foliation" of spacetime, usually there are many other ways, but the difficulty is usually in finding one way that works everywhere (it is always possible to do that separation only in some region of spacetime, but that exercise is not useful since you want to predict what happens everywhere, at any point in time).