I have been struggling some days now to understand the more general concept behind the abstract definition of geometry (probably the old-fashioned one due to Klein). As far as I know there are different kinds of geometries like euclidean, spherical, hyperbolic, elliptic, projective, affine etc. Although I've been looking around for the differences/similarities of them, didn't manage to sort it out clearly on my own, hence I think is a good place to ask here my query.
What I have so far is that projective geometry is a kind of extension of the Euclidean (and that does make sense in my head), and elliptic geometry is just the generalization of spherical in higher dimensions. But what's going on with the others? Is the projective geometry the general stage under which every other geometry can be built up? The Wiki article I think says for instance that projective space shares locally the same properties as models of spherical geometry, but don't really get that.. (as many many others unfortunately).
Frankly, I' m quite confused and a thorough answer behind the general philosophy of the aforementioned things (or probably a concise reference) it's really appreciated.
Also what can be said about the most recent definition of what Klein called geometry in his Erlangen Program?
I' m quite new here, so if I have done any mistake please feel free to remedy them! Thanks