From what I understand, a plane curve $\alpha: I \to \mathbb{R}^2$ is immersed in $\mathbb{R}^2$ if $\alpha'$ is everywhere injective.
An embedding of a plane curve $\alpha: \mathbb{I} \to \mathbb{R^2}$ would be a function $f: I \to \mathbb{R^2}$ such that $f$ is bijective, continuous and has a continuous inverse.
Is my understanding correct? Can anyone provide some examples or some motivation to the definitions? I'm a little lost here, and some clarification would be great.
An immersion is as you say a differentiable map $\alpha : I \to \mathbb{R}^2$ such that its derivative $\alpha'$ is injective at each point of the domain.
Why is this important? As far as I know, immersions become more relevant in the context of manifolds (of which curves are a special case). In general, if you have a map $f : M \to N$ between manifolds which is an immersion, it means the derivative $df_x : T_x M \to T_{f(x)}N$ is injective at each point $x \in M$. One reason this property is nice is that given an immersion $f$, for each $x \in M$ there is a neighborhood $U$ of $x$ in $M$ such that $f(U)$ is an immersion.
This does not mean that $f(M)$ in its entirety is a manifold (though this could happen). If $f :M \to N$ is an embedding, it means it is an immersion, and also a homeomorphism onto its image (in particular, injective). In this case, we get the full conclusion that $f(M)$ is a manifold.