Wolfram Alpha says:
An affine transformation is any transformation that preserves collinearity (i.e., all points lying on a line initially still lie on a line after transformation) and ratios of distances (e.g., the midpoint of a line segment remains the midpoint after transformation). In this sense, affine indicates a special class of projective transformations that do not move any objects from the affine space to the plane at infinity or conversely. An affine transformation is also called an affinity.
But I'm worried that this is inaccurate. If I have a horizontal vector of length 1 and a vertical vector of length 2 and I apply a non-uniform scaling to the space it's very easy to change the ratio to something other than 1:2.
The example they give of a midpoint of a line segment remaining the midpoint after transformation doesn't show the problem. Does the definition really mean "preserves ratios of distances between collinear points"?