What are positive and negative circles in weighted digraph?
A circle is a path in the graph ($v\to u_1 \to u_2 \to \cdots \to v$) starting from some node $v$, that goes back to node $v$, thus "closing a circle".

The "value" of a circle, is the sum of the weights of the edges in the circle. If they sum to a negative value -- the circle is "negative"; otherwise, it is "positive".