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A Geometry in which properties are preserved by Parallel Projection from one Plane to another.
In an affine geometry, the third and fourth of Euclid's Postulates become meaningless. This type of Geometry
was first studied by Euler. 
See also Absolute Geometry, Affine Complex Plane, Affine Connection, Affine Equation, Affine Group, Affine Hull, Affine Plane, Affine Space, Affine Transformation, Affinity, Ordered Geometry
References
Birkhoff, G. and Mac Lane, S. ``Affine Geometry.'' §9.13 in
A Survey of Modern Algebra, 3rd ed. New York: Macmillan, pp. 268-275, 1965.