What are a geometric system and a finite geometry? Wikipedia says

A finite geometry is any geometric system that has only a finite
  number of points.

I wonder what a geometric system is? Is it some set system $(E, F)$, where $E$ is a set and $F \subseteq \mathcal P(E)$ that satisfies some properties?
Or is a geometric system a metric space? I doubt it, because Wikipedia seems to say a projective place can be a finite geometry, and there is no metric on a projective place.
What is the definition of a finite geometry then? Is it some  set system $(E, F)$?
 A: There are lots of things that could be called geometric systems, so there isn't a single definition. I think this is meant to be interpreted as "a system of axioms" in the sense of synthetic geometry.
Most of those systems begin by telling you what the "points" are. Many more will then tell you what "lines" are, and probably some more axioms about how lines and points behave, and maybe they go on to make further assertions about parallels etc.
But the point is that nearly all of them tell you what points are, and as long as there are only finitely many points, and we don't mind what the rest of the axioms say. Having finitely many points is enough to call it a finite geometry, by this definition.
A: A wide variety of geometric systems can be defined, but generally there is not a metric space that is associated with the definition.
A very general setting of points in $E$ and "incidence sets" in $F$, which are collections of points in $E$, so each incidence ("block" or "line") is a subset of $E$, i.e. $F \subseteq \mathcal{P}(E)$.  One might generalize further to allow multisets of points, i.e. the same point appearing more than once in a block.
Some additional assumptions may be imposed to define special geometries.  In a projective space every pair of lines intersects in exactly one point, for example.  The Wikipedia article on finite geometries provides many links to such related topics.  There is some amount of conflicting terminology between combinatorial approaches and "geometric" analogies.
