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.

The first space we are introduced in our lives are euclidean spaces, which are the classical beginning point of classical geometry. In these spaces, there is a natural movement between points that are translations, i.e., you can move in a natural way from a point $p$ to a point $q$ through the vector that joint them $\overrightarrow{pq}$.
In this way, vectors represent translations in the euclidean space. Therefore vector spaces are the natural generalization of translations of spaces, but which spaces? Here is where affine spaces are important, because they recover the concept of points which the "arrows" (vectors) of a vector space move.
In conclusion, an affine space is mathematical modelling of an space of points whose main feature is that there is a set of preferred movements (called translations) that permits to go from any point to other point in an unique way and that are modeled through the concept of vector space. Or in other words, affine spaces represent the points that the arrows of vector spaces move.
