This is basically from Hartshorne Exercise I.5.13 where he writes there is a correspondence from the set of plane projective curves of degree $d$ to points in a projective space $\mathbb P^N$ where $N=\binom{d+2}{2}-1$. All seems reasonable and I can easily produce this canonical correspondence.
However he writes that this is not 1-1 if we look at the image of a curve of degree $d$ that is reducible. I seem to be too stupid to find two points in the preimage of a point in $\mathbb P^N$. Perhaps I misunderstood something?
$\newcommand\P{\mathbb P}$I think I know why I was not getting the example:
So Hartshorne meant a map from points in $\P^N$ to the set of curves that can be defined by forms of degree $d$. So not only reverse of what I thought, but also we are not talking about the degree of the curve here which I misunderstood as well. One way to get a fiber with more than one point using this map is to consider $d=4$ and now the point in $\P^N$ corresponding to $x^2y^2$ which is different from the point in $\P^N$ corresponding to $x^3y$ map to the same curve.