What are affine spaces for? I'm studying affine spaces but I can't understand what they are for. 
Could you explain them to me? Why are they important, and when are they used? Thanks a lot.

A vector space is an abstraction of how geometrical vectors (in the plane, say) behave. You can form linear combinations of vectors. Not all vectors look alike; they can vary in direction and in magnitude, and in particular there is the zero vector which is special. And so on...
An affine space is an abstraction of how geometrical points (in the plane, say) behave. All points look alike; there is no point which is special in any way. You can't add points. However, you can subtract points (giving a vector as the result). And you can form convex combinations of points. And so on...
