An affine combination is like a linear combination, however for coefficients $a_i$:
$$\forall a_i \in F: \sum_{i=0}^{n} a_i = 1$$
However you can also subtract points from each other in affine space, producing a distance vector in the process, but isn't this subtraction just an affine combination with $\sum_{i=0}^{n} a_i = 0$?
Like you say, the coefficients $a_i$ of an affine combination $\sum a_i x_i$ satisfy $\sum a_i = 1$, so with this definition it's a contradiction in terms to speak of an "affine combination with $\sum a_i = 0$". On the other hand, the condition $\sum a_i = 0$ is a linear condition on $n$-tuples of coefficients, so the space of $n$-tuples satisfying this condition is, as mentioned, a vector subspace of $\Bbb F^n$.
