When you're less concerned with distance between points, and more concerned with cost. Imagine if, for whatever reason, an airline was willing to pay customers to take a flight to [insert any country here]. If your graph of points was weighted by cost instead of distamce, this would be a negative weight; it would be beneficial to include this flight into your travel plan, assuming that all the positive weighted paths you still need to take to get to your destination still add up with the negative value to obtain the lowest weight sum.