What is Affine Transformation? What is affine transformation? Which distribution families are closed under affine transformation?
 A: An affine transformation has the form $f(x) = Ax + b$ where $A$ is a matrix and $b$ is a vector (of proper dimensions, obviously).
A: Affine transformation(left multiply a matrix), also called linear transformation(for more intuition please refer to this blog: A Geometrical Understanding of Matrices), is parallel preserving, and it only stretches, reflects, rotates(for example diagonal matrix or orthogonal matrix) or shears(matrix with off-diagonal elements) a vector(the same applies to many vectors/a matrix), and the "non-affine"(also a type of projective transformation as explained in the comment by @whuber) transformation may be like the first example in the following diagram: 

More generally speaking affine transformation has the following three properties: 

straight lines preserved
  parallel lines preserved
  ratios of lengths along lines preserved (midpoints preserved)

A: So I look here:  http://mathworld.wolfram.com/AffineTransformation.html
It is a rotation.  All points on a line, stay on the same line.
Per @Luca:
It can have scaling, shear, translation as well.  No bending.  Straight lines are always straight.  
