According to Guillemin and Pollack's Differential Topology:
The sum of the orientation numbers at the boundary points of any compact oriented one-manifold $X$ with boundary is zero.
By The Classification of One-manifold, every compact, connected, one-dimensional manifold with boundary is diffeomorphic to $[0,1]$ or $S^1$.
I think oriented manifold means a manifold with an orientation. By definition, an orientation of a manifold with boundary is a smooth choice of orientations for the tangent space. (By the way, does "smooth choice of orientation" mean the orientation varies smoothly? Then since orientation is either $0$ or $1$, it means orientation does not change?)
So I assume this means $X$ is connected? Hence, we can apply the classification, and $X$ must be diffeomorphic to $[0,1]$ or $S^1$. So for the $[0,1]$ case, its two ends just cancel out regardless the orientation of $X$; for $S^1$, the boundary is trivial - non exist, correct?
It is really frustrating that I have to work out pages of details for many of the one sentence claims made in this book. But sincere thanks to MathSEers, who essentially are teaching me this subject - without you I could have gave up at Page 10, at most.