The past of any physical event (object, system, etc) consists of everything that might (by the principles of causality) have potentially influenced that event; while its future consists of everything that it might potentially influence.
Let $(X,g,o)$ be a spacetime, that is a Lorentzian manifold $(X,g)$ equipped with time-orientation?. This time-orientation consists precisely of specification of which timelike and lightlike curves are future-directed (and which are complementarily past-directed).
Let $x$ be a point in this spacetime. Then:
the future of $x$ is the subset $J^+(x)$ of all points of $X$ connected to $x$ by a future-directed timelike or lightlike curve starting at $x$;
the past of $x$ is the subset $J^-(x)$ of all points of $X$ connected to $x$ by a future-directed timelike or lightlike curve ending at $x$.
Let $A$ be a more general subset of this spacetimes. Then:
the future of $A$ is the subset $J^+(A)$ defined as the union of $J^+(x)$ for all $x \in A$;
the past of $A$ is the subset $J^-(A)$ defined as the union of $J^-(x)$ for all $x \in A$.
A Cauchy surface $\Sigma$ in $(X,g)$ is a minimal subset of $X$ with the property that $X$ is the union of the future and past of $\Sigma$. (Does this suffice to define Cauchy surfaces in the case of a Lorentzian manifold that admits a time-orientation?)
The operations $J^+$ and $J^-$ are (separately) Moore closures on the power set of $X$. Stated explicitly (for $J^+$):
Sometimes one wants to remove $x$ itself from $J^+(x)$ and $J^-(x)$ (or more precisely, to include $x$ only in the case of a closed timelike curve through $x$). However, the operations $J^+$ and $J^-$ are not quite as mathematically well-behaved in this case. (Note that $J^+(A)$ may still intersect $A$, or even contain all of $A$, even in the absence of closed timelike curves.)