Definition of "immersed plane curve" The exact meaning of immersed plane curve is not clear to me and I would like to request some help with clarifying it here:
A plane curve is a map $f: \mathbb R \to \mathbb R^2$. For example, the circle $f(t) = (\cos t, \sin t)$ is a plane curve. 
An immersion is a (smooth) map between (smooth) manifolds $f: M \to N$ such that the pushforward (=Jacobian matrix) is injective at every point in the domain. 
I assumed that an immersed plane curve  must be a plane curve that is an immersion. But the Jacobian of a map $f(t) = (x(t),y(t))$ is given by $(x'(t) \  \  y'(t))$ and that, of course, is not going to have non-zero determinant for any $t$. 

What am I doing wrong here?


Your "of course" is wrong; for example the curve $f(t)=(t,t^3)$ has derivative at $t=0$ that is not injective, namely the matrix $[0,0]$.

I'd misread your question - I thought you were asking if every plane curve was immersed, but now I see you were asking why any are.
Remember that determinants only make sense for square matrices, which I should have corrected you on earlier anyway - the relevant issue (the pushforward being injective) is equivalent to the matrix having full rank. For example, at $t=1$, the curve $f(t)=(t,t^3)$ has derivative $[1,3]$, which has rank $1=\min\{1,2\}$ (i.e., "full rank"). Equivalently, as a map $$\mathbb{R}\cong T_1\mathbb{R}\longrightarrow T_{(1,1)}\mathbb{R}^2\cong\mathbb{R}^2$$
it sends $s$ to $(s,3s)$, which is certainly injective.
