The notation you give is fairly common, and Jasper explains its meaning. If you wanted to write the same sum in a more explicit (albeit heavy-handed) way, you could write something like $ b_i = \sum_{j \in J_i} A_{ij}x_j $ where $J_i = \{ j \in \mathbb{N} \mid 1 \leq j \leq n,\ j \neq i\}$. This notation means to sum over every value of $j$ belonging to the set $J_i$. The subscript $i$ is a way to denote that you will have a different "$J$" set for each choice of $i$. In other words, each "$J$" set depends on $i$.
As remarked in the comments, there are ways to denote this sum that are easier on the eye. The point of my answer is that every summation is just a sum over some indexing set. If all else fails, you can always say "sum over indices belonging to some set" and then describe to the reader what that set is.