The map $\varphi : M \to N$ is an embedding if $\varphi$ is a homeomorphism, an immersion, and $\varphi(M) \subset N$, where $M$ and $N$ are manifolds.
The curve $\alpha(t)=(t^3-4t,t^2-4)$ has a self-intersection for $t=2$ and $t=-2$ (because $\alpha(2)=\alpha(-2)=(0,0)$).
Why is this curve not an embedding? I do not know how $\alpha(\mathbb R) \subset \mathbb R^2$ does not hold. Because the curve has a self-intersection, there's some $t$ such that $\alpha(t) \not\in \mathbb R^2$?
In general you want the map to be at least injective. Also the image to be smooth could be a good thing to ask. Even tho $$\alpha'(t)=(3t^2-4,2t) \not= (0,0)$$ unfortunatley the map is not injective because $\alpha(2)=\alpha(-2)$.
Loosely speaking what you really want is something like a clone of $\mathbb R$ in $\mathbb R^2$, then of course you don't want self intersection and you want it to be smooth. A general way to ask for this is to ask for injectivity and for the injectivity of the induced maps (one per point) between the tangent spaces.