How many times the parabola $y=x^2$ intersects the origin? The answer of this question is 1, right? but I'm about to study algebraic curves by this book
and I was surprised by this theorem:

If I'm right the immediate  corollary of that is $y=x^2$ intersects the origin two times.
Then he continues given an example:


I don't understand this, anyone can help me?
Thanks a lot

This is an instance where higher generality and scheme theory gives a more satisfactory description.
What is the intersection of two algebraic varieties? For simplicity, let both be affine subvarieties of the same affine space. For e.g., let the first curve be given in the affine plane by the vanishing ideal $I_1$ generated by the polynomial $y-x^2$. Let the second curve be given in the affine plane by the vanishing of ideal $I_2$  generated by the polynomial $y$.
Roughly, our algebraic varieties are zero sets of certain polynomials. So, the intersection of two of them should be the common zero set of all the polynomials in the ideals corresponding to either variety, i.e., in our case, it will be the zero set of the ideal $(I_1, I_2)$ generated by $I_1$ and $I_2$.
Which, unfortunately, is not an algebraic variety. This would have been an algebraic variety if the intersection were transversal, i.e., it were not tangential. The intersection would be a single point in that case, which is not quite the case for tangential intersection. There things are much more clear if we generalize to schemes. Consider the two relations $y=0$ and $y=x^2$. Taken together, this means the presence of an element $[\epsilon]$ in the structure sheaf such that $\epsilon ^2=0$ and this is an infinitesimal signifying in a "thick point" at the intersection. This POV is very clearly explained in Mumford's Red Book, which is a very good introductory  source for learning Varieties and Schemes at once. A book more at the level of the one you cite is Smith, Kahanpää, Kekäläinen and Traves, in which this is explained more informally.
