In p.256 of Loring Tu's An Introduction to Manifolds, there is an example:
Example. Suppose $c:[a,b]\to M$ is a $C^\infty$ immersion whose image is a 1-dimensional manifold $C$ with boundary. An orientation on $[a,b]$ induces an orientation on $C$ via the differential $c_{*,p}:T_p[a,b] \to T_{c(p)}C $ at each point $p\in [a,b]$.
How does the second sentence hold? An orientation of a manifold is by definition a global frame that is locally represented by a continuous frame, so to show that $C$ admits an orientation, we have to show that there is such a frame $C \to TC$, but I don't see how $c$ induces such a frame.
First, one needs the correct definition of orientation for a (smooth) $n$-dimensional manifold: This can be defined in many ways, most common are:
Choice of an orientation atlas, i.e. an atlas with orientation-preserving transition maps.
Choice of a nowhere vanishing section of $\Lambda^n TM$, equivalently, of a nowhere vanishing $n$-form on $M$.
Whatever definition you use, the following holds (I will leave you a proof as an exercise):
Lemma 1. Suppose that $M, N$ are smooth oriented $n$-dimensional manifolds, $M$ is connected and $f: M\to N$ is a local diffeomorphism. Then $f$ preserves orientation at one point if and only if $f$ preserves orientation everywhere.
Now, consider the case when $n=1$. Every connected $1$-dimensional manifold is orientable and has exactly two different orientations (that's another exercise you should do). Therefore, if $f: A\to C$ is a local diffeomorphism between connected 1-dimensional manifolds, and $A$ is oriented, then $f$ determines an orientation on $C$ as follows: Pick one point $x\in A$ and choose one of two orientations on $C$ (call it $o_C$) such that $f$ preserves orientations at $x$. Then, by Lemma 1, $f$ preserves orientation everywhere. Thus, the orientation $o_C$ does not depend on the choice of $x$. This is the orientation on $C$ determined by $f$.
Now, apply this to your situation when $A=[a,b]$, $c: A\to C$ is an immersion. (In your example, $A$ has boundary, but this is irrelevant, just replace $[a,b]$ with $(a,b)$.)
Lastly, in general, if $M$ and $N$ are connected manifolds of the same dimension, $f: M\to N$ is a surjective local diffeomorphism, then $f$ does not determine an orientation on $N$, since $N$ need not be orientable.