The only operations defined on points in an affine space are
- point-vector addition. this yields a new point.
- point-point subtraction. this yields a vector.
This can be extended to an affine sum, $\sum_i \alpha_i P_i$, where $\sum_i\alpha_i = 1$.
However, it is possible to derive an expression using these operations that should otherwise be undefined.
Imagine points, $P$, $Q$, $R$, and $S$ such that
$ P - Q = R - S $
This leads to
$ P + S = Q + R$
Each side of the equation is now a point-point addition which is not defined. It is also not an affine sum.
How should each side of this equation be interpreted?
This leads to weighted points in affine space. The weight of a point must be nonzero and usual affine points have weight one by definition. Given weighted points $\,aP\,$ and $\,b\,Q\,$ their sum is $\,aP+b\,Q\,$ which has weight $\,c:=a+b\,.$ If $\,c\,$ is nonzero then this is the weighted point $\,c\frac{aP+b\,Q}c.\,$ If $\,c=0\,$ then the sum is the vector $\,a(P-Q).\,$ In general a "point" with weight $0$ is just a vector of the underlying vector space. That is, all weighted points have nonzero weights, but if a sum of weighted points has weight zero, then it becomes a vector instead.
I give more details in my answer to MSE question 1059220 "what is the difference between linear transformation and affine transformation?".