I have found the following statement in one of my exercises:
Let $W$ be a oriented 1-manifold with boundary. Then its boundary has "as many positive points as negative points. I don't understand this statement, what are "positive" or "negative" points here.
As you consider a 1-manifold, it is a disjoint union of circles and intervals, and the only connected components with boundary are closed intervals. Each of them has an orientation, you can think this as a direction, or an arrow telling you where is left and where is right. A positive point of the boundary is a point on the very right side, and a negative point of the boundary is a point that is on the very left of the interval.
Edit: However, I think you want the manifold to be compact, as else $\mathbb{R}^+$ would be an obvious counterexample.