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.
Orientation has several possible, equivalent, definitions. One for smooth manifolds is a choice of ordered basis of the tangent space for positive dimension, while a point gets a $\pm 1.$
For $\mathbb S^1$ a novanishing tangent vector field along the circle supplies an orientation. For the closed unit interval, an arrow along the segment orients the open segment, then the enpoint at which those arrows point gets orientation $+1,$ the other endpoint gets $-1.$