What structures does "geometry" assume on the set under study? The Wikipedia's article for geometry is somehow overwhelming. To make things clear, allow me to ask some questions:

*

*I wonder if "geometry" can be defined as the study of a metric
space (possibly with or without other structures)?


*Any thing more general than metric space (such as uniform spaces and topological spaces) is not in the scope of "geometry"?


*Does "geometry" assume the set under study to have some algebraic
structure?
Also there is algebraic geometries.


*Is the underlying set a topological vector space, normed space,
inner product space, or even Euclidean space?


*Since projective and affine spaces are pure algebraic concepts
without metrics, why are there "projective geometry" and "affine geometry"?
Thanks and regards!
 A: A partial (i.e. highly incomplete) answer to your questions on algebraic geometry:

Is the underlying set a topological vector space, normed space, inner product space, or even Euclidean space?

It depends - in a lot of algebraic geometry, you think about particular subsets of affine or projective spaces, which are not equipped with standard metrics, norms or inner products. In more abstract settings you might even be dealing with schemes, which are a big generalisation of these spaces. Very loosely, they can be thought of as being constructed by turning rings into geometric objects and then gluing them together. For example, $n$-dimensional affine space over a field $K$ is the geometric object associated to the polynomial ring $K[x_1,\ldots,x_n]$, and projective space can be obtained by gluing together some affine spaces in a particular way.
It is definitely too strong to insist that the underlying space is Euclidean, because that would be ignoring all of the interesting geometry on spheres and hyperbolic spaces, among others. (Here I mean geometry in the more traditional sense of theorems about distances, angles, intersections of lines and so on, but there is a lot more to spherical and hyperbolic geometry as well).

Since projective and affine spaces are pure algebraic concepts without metrics, why are there "projective geometry" and "affine geometry"?

I may have got the wrong idea from your question here, but I've taken it to be asking what kinds of geometric theorems you can have without metrics. A number of the big theorems in algebraic geometry are primarily algebraic in nature, but are motivated by geometric questions, while others are more recognizable as being geometric theorems.
One example is as follows. The main object of study in affine and projective algebraic geometry is an algebraic variety, which is a subset of affine or projective space defined by an ideal of polynomial equations. In particular, any curve in the plane ($2$-dimensional affine space) defined by a single polynomial is an algebraic variety. Up to worrying about degenerate cases, we can say that the degree of the curve is the degree of the polynomial defining it. Then Bézout's theorem states that two "general" curves of degrees $d_1$ and $d_2$ intersect in $d_1d_2$ points (there is a lot of interesting mathematics involved in saying exactly what "general" means). So this is an example of a fairly strong theorem, which says a lot about how curves behave, without any reference to a metric.
A: I would say that "geometry" in a strict sense needs a notion of length AND some kind of inner product (I'd say a quadratic form). Thus the study of most manifolds is geometry since it's usually equipped with a standard Riemanian or Lorentz form (it's the case in particular for projective or affine spaces on $\mathbb{C}$ or $\mathbb{R})$. 
I wouldn't say that the study of general, abstract metric spaces is "geometry" in that sense (and therefore much less so for general topological spaces), but I think it's not something set in stone. 
Also there is a very broad acceptation of the term "geometry" as in "algebra, analysis, probabilities and geometry", and since general topology fits better in geometry than in analysis or algebra, you could say that it belongs there, but for me it's a slight abuse of language. 
