What are affine spaces for?

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...

Regarding "when would I use it" -- affine space is the natural setting for computer-aided design, computer-aided manufacturing, and other computer applications of geometry. People who develop software in this field all know that you have to carefully distinguish points and vectors (even though they might both be represented as triples of numbers), and avoid "illegal" operations like adding two points. The more mathematically inclined members of the community understand that they are working in an affine space.

So, like most abstractions, affine spaces may or may not be helpful to you, depending on how your brain works. But the particular affine space $R^3$ is very important. As celtschk pointed out, it's the space we live in. And, in particular, it's the space we compute in.

The euclidean space $E$ of high school geometry (2d or 3d) is an affine space, but one with extra structure: You can measure lengths and angles; among the angles there are distinguished ones, namely right angles, and among the ellipses there are distinguished ones, namely circles.

The affine structure of $E$ is what remains when you throw away compass, set-square and protractor. Parallelity is still recognized. The allowed tools left are the ruler and a device to determine the ratio of lengths on parallel lines.

A theorem of affine geometry in the plane is the following: Assume that the lines $a$ and $b$ intersect in a point $P$, that $A_1$, $A_2\in a$, $\ B_1$, $B_2\in b$, and that ${\rm vec}(PA_1)={\rm vec}(A_1A_2)$, $\,{\rm vec}(PB_1)={\rm vec}(B_1B_2)$. Then the lines $A_1\vee B_1$ and $A_2\vee B_2$ are parallel.