In Rienmannian geometry space can curve at different places (see manifolds) here we look at geometries where the curve of space is constant.
flat space 

Hyperbolic Geometry space curves outward 

Spherical Geometry and Elliptic Geometry space curves inward 
Terminology
Here we look at the terminology such as geometries, spaces, models, projections and transforms. Its quite difficult when we start dealing with nonEuclidean geometries because we use similar terminology that we are used to in conventional Euclidean space but the terms can have slightly different properties. For example, the concept of a 'line' can look different in different geometries:
Geometries and Spaces
Rienmannian geometry defines spaces generally in terms of manifolds, here we are interested in homogeneous, isotropic spaces which have no preferred points or directions, examples are:
parallel postulate  number of unique parallel lines through point  space curves  Inventors  Point at Infinity  Distance Measure  

Eulidean Geometry  1  none  Euclid  where parallel lines meet  √(x² + y²) 
Hyperbolic Geometry  ∞  space curves outward so lines dont meet  Lobachevskii and Bolyai 

Elliptic Geometry 
0  space curves inward so all lines meet 
Models
Examples are:
 Upper halfplane model
 Poincare disc model
 Projective model
 Conformal model
Projections
Transforms
Invariant
Examples are:
 Möbius Transform
 Lorentz Transform
Parallel Postulate
If we take away the parallel postulate from Euclidean Space.
This leads to:
 parallel postulate is false
 the angles of a triangle do not add to π
 for a shape of a given size, there does not in general, exist a similar shape of a larger size.
Possible mappings between Euclidean Space and Hyperbolic Space:
properties:
projective  conformal (Poincaré disc )  

straight lines  geodesics  segments of circles  
angle  preserved  
internal angles of triangle  π(α+β+γ)=CΔ  
distance between two points  ln  