Is Euclidean Geometry complete and unique Please help me understand this concept of completeness of geometry and set me on the right path. 

This is my context: From wikipedia, a formal system is complete if every tautology is also a theorem. For me this means that a system of axioms is complete if every statement that is true can be proved from the axioms (the question is tautologies in what kind of logic, first order, second order,...?). Anyway, it seems then that the concept of completeness depend on the system of axioms. 

So, in the case of Euclidean Geometry, its completeness depends on their axioms (For example Euclid's Axioms, Hilbert Axioms, Tarki's Axioms,etc). Comparing for example the axioms of Hilbert and the axioms given by Tarski, I can see that they are essentially different in that Hilbert uses second order logic and Tarski's only first order logic. Also Tarski proved that his system of axioms is complete. 

Now, the incompleteness theorem of Gödel stated that the system of axioms of arithmetic is incomplete. So, when we consider $R^2$ and $R^3$ as a representation of Euclidean Geometry we are talking about a system that is not complete because we are using the real numbers. So, are we speaking of different Euclidean Geometries? I'm totally confused.
It sounds like you might get a lot out of Greenberg's short paper Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries.

In it he discusses how the ordinary theory of plane geometry is incomplete (pg 202), but why Tarski's geometry is complete (section 3.3, page 215).
For any fragment of euclidean geometry seems to be topologically and notionally equal.  R2 and R3 are different only in dimension.  

One should be careful about what is being implied by Euclid's geometry.

Just as the curvature of space (the sum of angles of a triangle for example), leads to euclidean, hyperbolic and spherical geometries, there is a second statement for which any two, but not all three parts, can be true.  These are set in italics.

In a complete  and orientable space, lines cross once.

Euclid's geometry deals with a fragment of the plane, and never considers completeness.

The spheric geometries are necessarily complete, so one either rejects orientable to give elliptic, or once, giving spherical geometry.

So when one starts talking of the nature of the euclidean geometry and the horizon at infinity, one gets a presentation as a line (projective) or point (inversion).  Some people consider what happens on the other side of infinity, that is, they try to complet the euclidean plane.

A complete euclidean plane is perfectly modelled by the sphere at infinity.  This is the centre of a planes through a point, and thus must be a the centre of a plane centred on a real point.  

The actual geometry is called "Möbius" geometry, but corresponds to the implemetation of 0/0 as undefined, that any circle is a straight line, and crosses other circles twice.  If one supposes that 'any point on the surface' is 'a point at infinity', and that straight lines contains the point at infinity, this folds down to the inversion model of the euclidean plane.  

In short, Möbius geometry is oE, or orientable-complete-euclidean, whereas Euclid studied the geometry of orientable-fragment-euclidean.  

Hyperbolic geometry is necessarily a fragment of space, even at 'infinity'.  This is because there is no meaning for a plane in 3D centred on a real point, unless you get Meta-infinities come to hand.