what are the "coordinates" of an affine variety? Let $X$ be an affine variety of dimension $d$ inside the affine space $\mathbb{A}^n$. What do we mean when we say "let $u_1,\dots,u_d$ be coordinates of $X$"? Does this mean that we can describe every element of $X$ by a $d$-tuple? If yes, how can we see that?
This terminology appears for example in the solution of problem I.3.15(d) in Hartshorne here: http://www.math.northwestern.edu/~jcutrone/Work/Hartshorne%20Algebraic%20Geometry%20Solutions.pdf
 A: One says $u_1,\ldots, u_d$ are coordinates on $\mathbf{X}$ if they generate the coordinate ring $k[X_1,\ldots, X_n]/I(\mathbf{X}).$
E.g. If $\mathbf{X}=V(Y-X^2) \subset \mathbb{A}^2$ then the coordinate ring is $k[X,Y]/(Y-X^2)$ which is generated by $x:= X + (Y-X^2)$ so $\{ x \}$ is a set of coordinates for the parabola. The plane $Z=0$ inside $k^3$ has a coordinate ring generated by the images of $X$ and $Y,$ so those are coordinates for that plane. 
If values the $\{u_i\}$ are specified, then this determines a value for each $X_i$ because each element $X_i+I(\mathbf{X})$ is generated by the coordinates. The n-tuple $(X_1,\ldots, X_n)$ lies on $\mathbf{X}.$ Note that in the special cases where these generators are also algebrically independent (such as when $\mathbf{X}$ is rational), specifying $X_i$ on $\mathbf{X}$ determines the $u_i$ as well.
Back to the parabola example: Specifying $x=a$ gives $X=a, Y=a^2,$ and specifying a point on the parabola determines $x.$ 
You may also want to look at the top of page 100 of Vakil's notes and read around accordingly. 
