I'm having trouble understanding the following example from Vakil's "The Rising Sea":
For example, consider a map from the parabola in $\mathbb{C}^2$ (with coordinates $a$ and $b$) given by $b=a^2$, to the "curve" in $\mathbb{C}^3$ (with coordinates $x,\,y$ and $z$) cut out by the equations $y=x^2$ and $z=y^2$. Suppose the map sends the point $(a,b)\in\mathbb{C}^2$ to the point $(a,b,b^2)\in\mathbb{C}^3$. In our new language, we have a map
$\operatorname{Spec} \mathbb{C}[a,b]/(b-a^2)\to \operatorname{Spec} \mathbb{C}[x,y,z]/(y-x^2,z-y^2)$ given by $\mathbb{C}[x,y,z]/(y-x^2,z-y^2)\to\mathbb{C}[a,b]/(b-a^2),\,(x,y,z)\mapsto(a,b,b^2)$, i.e. $x\mapsto a,\,y\mapsto b$ and $z\mapsto b^2$.
My problem is that $\mathbb{C}[a,b]/(b-a^2)$ should only have two "coordinates" and not three, as far as I understand. Where lies my mistake?
There are two types of maps involved in Vakil's example, and I think your confusion stems from the notation seemingly mixing up these two.
The first is a geometrical map, which could be thought of as a function $\overline{\phi}:\mathbb{C}^2\to\mathbb{C}^3$, and are specified by three coordinate functions, say $\overline{\phi}_x, \overline{\phi}_y, \overline{\phi}_z: \mathbb{C}^2\to \mathbb{C}$, each taking two complex numbers as input and spits out a complex number. Specifically they are $\overline{\phi}_x(a,b) = a$, $\overline{\phi}_y(a,b)=b$, and $\overline{\phi}_z(a,b)=b^2$. Note that this function does have three coordinates as you expected. $\overline{\phi}$ can also be restricted to a map from the curve cut out by $y = x^2$ on $\mathbb{C}^2$ to the curve cut out by $y=x^2$ and $z=y^2$ on $\mathbb{C}^3$.
The second is an algebraic map, which is a homomorphism of algebras $\phi:\mathbb{C}[x,y,z]\to \mathbb{C}[a,b]$. Such maps are specified by the images of the generators $x$, $y$, and $z$. In other words, each polynomial $p(x,y,z)\in\mathbb{C}[x,y,z]$ is mapped to $p(\phi(x),\phi(y),\phi(z))\in\mathbb{C}[a,b]$, where $\phi(x), \phi(y),\phi(z)$ are elements of the ring $\mathbb{C}[a,b]$, in this case the polynomials $a$, $b$, and $b^2$ respectively. To define this map you need to make three specifications, because there are three generators of $\mathbb{C}[x,y,z]$, and we don't need to care about the number of generators of $\mathbb{C}[a,b]$. This map also induces a homomorphism of the quotient rings $\mathbb{C}[x,y,z]/(y-x^2,z-y^2)\to\mathbb{C}[a,b]/(a-b^2)$, because the ideal $(y-x^2, z-y^2)$ gets mapped into the ideal $(a-b^2)$.
I agree that the notation used by Vakil is somewhat confusing. The tuple notation seems at first glance to indicate that somehow elements of $\mathbb{C}[x,y,z]$ or $\mathbb{C}[x,y,z]/(y-x^2,z-y^2)$ have "coordinates" $(x,y,z)$, but that is an unfortunate coincidence. The elements of these rings are polynomials in $x$, $y$, and $z$, and the shorthand $(x,y,z)\mapsto (a,b,b^2)$ is only meant to indicate what the image of the generators are. Also notice that the arrow is "going in the wrong direction", so it cannot be interpreted as the desired geometrical map.