Well, to quote from Wolfram MathWorld directly,
Given an affine variety $V$ in the $n$-dimensional affine space $K^n$, where $K$ is an algebraically closed field, the coordinate ring of $V$ is the quotient ring $K[V] = K[x_1 , \dots , x_n] / I(V)$.
My question is simply this, why is it called the "coordinate ring"? In what sense does this ring give or can be said to define coordinates on the affine variety?
A lot of algebraic geometry is developed by analogy with differential geometry.
In differential geometry, we speak of manifolds rather than varieties. The main feature of the definition of a manifold is that it is covered by coordinate charts: open sets $U$ together with a (smooth) homeomorphism $x : U \to \mathbb{R}^n$ (or alternatively, to an open subset of $\mathbb{R}^n$). The individual components of $x$ are called coordinates.
As an example, consider the pair of functions $(x,y)$ defined on the Euclidean plane after choosing coordinate axes.
The situation here is not too different. On $\mathbb{A}^n_K$, affine $n$-space over a field $K$, the standard 'coordinate chart' has coordinates given by the $n$ variables. Since we're doing algebra, the 'functions' we can make out of these are polynomials.
The term "coordinate function" here refers to a function we can build out of coordinates — much like the ring of smooth continuous real-valued functions on a manifold. The coordinate ring is the ring of all such functions. (although, I don't think the phrase "coordinate function" tends to be used this way in differential geometry)
For subvarieties $V \subseteq \mathbb{A}^n_K$, we understand coordinate functions to be restrictions of coordinate functions on $\mathbb{A}^n_K$, thus we use the quotient ring of the coordinate ring of $\mathbb{A}^n_K$.
Furthermore, since every element of $K[V]$ can be viewed as the image of $t$ under a unique $K$-homomorphism $K[t] \to K[V]$, we can actually identify coordinate functions on $V$ with algebraic maps $V \to \mathbb{A}^1_K$.
(if we're doing absolute geometry, the target would instead be $\mathbb{A}^1_\mathbb{Z} = \operatorname{Spec} \mathbb{Z}[t]$)
The $i$th "coordinate function" on $K^n$ is the function $K^n\to K$ taking $(x_1,\dots,x_n)$ to $x_i$, its $i$th coordinate. Thinking of elements of $K[x_1,\dots,x_n]$ as functions on $K^n$ in the usual way, $x_i$ is the $i$th coordinate function. The ring $K[V]$ is then the $K$-algebra of functions on $V$ generated by the restrictions of all of these coordinate functions to $V$. So, as the algebra generated by the coordinate functions, it is the "coordinate ring".
